Study to Assess Safety and Activity of Combination Therapy of VRC07-523LS and Vorinostat on HIV-infected Persons

Inclusion Criteria: 1. ≥ 18 years and \< 65 years of age 2. Ability and willingness of participant to give written informed consent. Note: Due to the lack of foreseeable benefit to study participants, mentally incompetent participants will not be enrolled. 3. HIV infection documented by any licensed rapid HIV test or HIV enzyme or chemiluminescence immunoassay (E/CIA) test kit at any time prior to study entry and confirmed by a licensed Western blot or a second antibody test by a method other than the initial rapid HIV and/or E/CIA, or by HIV-1 antigen, plasma HIV-1 RNA viral assay. A reactive initial rapid test should be confirmed by either another type of rapid assay or an E/CIA that is based on a different antigen preparation and/or different test principle (e.g., indirect versus competitive), or a Western blot or a plasma HIV-1 RNA viral load. 4. On continuous antiretroviral therapy (ART defined below under Inclusion Criterion #5) for at least 24 months prior to screening. Note: Continuous ART prior to screening is defined as not missing more than 9 total days in the 3 months prior to screening. 5. Permitted regimens include: 1. At least 3 ART agents (not counting ritonavir if less than a 200 mg total daily dose or cobicistat as one of the agents) Note: One of the agents must include an integrase inhibitor, NNRTI (Non-Nucleoside Reverse Transcriptase Inhibitors), or a boosted-PI (protease inhibitor). OR 2. Two ART agents in which one of the agents is either a boosted PI or an integrase inhibitor. Note: Other fully suppressive antiretroviral combinations will be considered on a case-by-case basis. Note: Prior changes in, or elimination of, medications for easier dosing schedule, intolerance, toxicity, improved side effect profile or within a drug class are permitted if an alternative suppressive regimen was maintained but not within 30 days prior to screening. Note: Changes in drug formulation or dose are allowed (e.g., tenofovir disoproxil fumarate (TDF) to tenofovir alafenamide (TAF), ritonavir to cobicistat, or separate ART agent dosing to fixed-dose combination), but none within 30 days prior to screening. 6. Ability and willingness of participant to continue ART throughout the study. 7. Able and willing to adhere to protocol therapy, schedule, and is judged adherent to antiretroviral therapy. 8. Plasma HIV-1 RNA \<50 copies/mL at 3 time points in the previous 24 months prior to screening and never ≥50 copies/mL on 2 consecutive time points in the last 24 months. Note: The documented plasma HIV-1 RNA must be performed by any US laboratory that has a Clinical Laboratory Improvement Amendments (CLIA) certification or its equivalent. 9. At least 1 documented plasma HIV-1 RNA result \<50 copies/mL ≥24 months but ≤ 36 months prior to screening. 10. Plasma HIV-1 RNA level \<50 copies/mL on an FDA-approved HIV RNA assay at screening, performed at US CLIA Certified Laboratory (or its equivalent). 11. CD4 cell count ≥ 350 cells/mm³ obtained within 90 days prior to study entry, performed at any US CLIA Certified Laboratory (or its equivalent). 12. Hepatitis C (HCV) antibody negative result within 60 days prior to study entry or, if the participant is HCV antibody positive, a negative HCV RNA within 60 days prior to study entry . 13. Hepatitis B surface antigen (HBsAg) negative within 60 days prior to study entry. 14. Interferon-gamma release assay (IGRA) for tuberculosis (TB) with negative results within 60 days prior to study entry. Note: Participants with a prior positive TB IGRA and documented evidence of completed prophylaxis treatment may enroll in the study and do not need to undergo IGRA at screening. Participants with a prior positive IGRA who have not completed prophylaxis treatment will be excluded. 15. Men and women who are not of reproductive potential (see below) are eligible without requiring the use of contraceptives. Acceptable documentation of sterilization and menopause is specified below. a) Written or oral documentation communicated by clinician or clinician's staff of one of the following: 1. Physician report/letter 2. Operative report or other source documentation in the patient record (a laboratory report of azoospermia is required to document successful vasectomy in any partner assigned male sex at birth, hysterectomy, oophorectomy, non-surgical permanent sterilization, or tubal ligation.) 3. Discharge summary 4. Documented or participant-reported absence of a period for ≥ one year must be confirmed with Follicle stimulating hormone-release factor (FSH) measurement elevated into the menopausal range as established by the reporting laboratory. 16. All participants must agree not to participate in a conception process (e.g., active attempt to become pregnant or to impregnate, sperm donation, in vitro fertilization, egg donation) while on study and for 4 months after their last infusion. 17. All men participating in sexual activity that could lead to pregnancy must agree to consistently use at least one of the following forms of birth control for at least 21 days prior to Visit 3 and for 4 months after their last infusion: a) Condoms (male or female) with or without a spermicidal agent b) Diaphragm or cervical cap with spermicide c) Intrauterine device (IUD) d) Tubal ligation e) Hormone-based contraceptive f) Successful vasectomy Note: For female partners who are receiving ritonavir or cobicistat, estrogen-based contraceptives are not reliable and an alternative method should be suggested. 18. Ability and willingness to provide adequate locator information. 19. Ability and willingness to communicate effectively with study personnel; considered reliable, willing, and cooperative in terms of compliance with the protocol requirements. 20. Adequate vascular access for infusion and leukapheresis. 21. Able to swallow pills without difficulty. 22. Agrees not to enroll on another study of an investigational research agent during the study period. Note: Investigational research agent is defined as any unlicensed investigational drug not yet approved by the FDA for intended use in humans. 23. Adequate organ function as indicated by the following laboratory values: Hematological: Absolute neutrophil count (ANC) ≥1,500 /mm\^3; Platelets ≥125,000 / mm\^3; Hemoglobin ≥ 13 g/dL (male) and ≥ 11 g/dL (females) Coagulation: Prothrombin Time or International Normalized Ratio (INR) ≤1.5x upper limit of normal (ULN) Chemistries: K+ levels - Within normal limits; Mg++ levels ≥ 1.2 milliequivalents/L but \<1.5 x ULN; Glucose - Screening serum glucose ≤ Grade 1 (fasting or non- fasting); Albumin ≥ 3.3 g/dL Renal: Creatinine clearance determined by the chronic kidney disease (CKD)-Epi equation Hepatic: Serum total bilirubin - Total bilirubin \< 1.5 X ULN. If total bilirubin is elevated, direct bilirubin must be \< 2 times the ULN range. Note: If participant is on an atazanavir-containing therapy, then a direct bilirubin should be measured instead of the total bilirubin and must be ≤ 1.0 mg/dL.; aspartate aminotransferase (AST, SGOT) and alanine transaminase (ALT, SGPT) ≤ 1.25 X ULN; Alkaline Phosphatase ≤ 2.0 X ULN; Lipase \< 1.6 X ULN; Urinalysis: Urine Protein - Negative or trace allowed ULN = upper limit of normal Exclusion Criteria: 1. Known allergy or sensitivity to components of VOR 2. Serious adverse reactions to VRC07-523LS formulation components, VRC01 or VRC01LS, including history of anaphylaxis and related symptoms such as hives, respiratory difficulties, angioedema, and/or abdominal pain. 3. Women without documentation of an FSH level indicating menopause, hysterectomy or bilateral oophorectomy, bilateral tubal ligation, or non-surgical sterilization. 4. Receipt of compounds with histone deacetylase (HDAC) inhibitor-like activity, such as valproic acid within 30 days prior to screening. Potential participants may screen after a 30-day washout period. 5. Any investigational research agent within 30 days before study entry. Note: Co-enrollment in observational only studies is permitted. Note: Co-enrollment in other studies using FDA approved medication that are not otherwise listed as prohibited will be evaluated by the study PI and permitted on a case by case basis. 6. Plasma HIV RNA ≥150 copies/mL in the 6 months prior to screening. 7. Weight \> 115 kg 8. Untreated syphilis infection (defined as a positive rapid plasma reagin (RPR) without clear documentation of treatment). Note: In cases of untreated syphilis, participant may rescreen following documentation of adequate treatment of syphilis 9. Current treatment for HCV with antiviral therapy or participants who have received HCV treatment within 6 months prior to screening. 10. Use of any of the following within 90 days prior to entry: immunosuppressive, immunomodulatory, cytokine, or growth stimulating factors such as systemic corticosteroids, cyclosporine, methotrexate, azathioprine, anti-CD25 antibody, interferon (IFN), interleukin-2 (IL-2). Not Exclusionary: \[1\] corticosteroid nasal spray; \[2\] inhaled corticosteroids; \[3\] topical steroids for mild, uncomplicated dermatitis; or \[4\] a single course of oral /parental prednisone or equivalent at doses \<2mg/kg/day and length of therapy \<11 days with completion at least 30 days prior to enrollment. 11. Current use of Coumadin, warfarin, or other Coumadin derivative anticoagulants. 12. Prior receipt of HIV immunotherapy within 6 months prior to screening. 13. Prior receipt of more than three doses of Vorinostat. 14. Prior receipt of humanized or human monoclonal antibodies (mAbs) within 2 months prior to screening, whether licensed or investigational, will have eligibility determined by the study PI on a case-by-case basis. 15. Received any infusion blood product, immune globulin, or hematopoietic growth factors within 90 days prior to study entry. 16. Pregnancy or breast-feeding. 17. History or other evidence of severe illness, malignancy, immunodeficiency other than HIV, or any other condition that would make the participant unsuitable for the study in the opinion of the investigator, for at least 90 days prior to screening. 18. History of autoimmune disease Not exclusionary: Persons with mild, stable, and uncomplicated autoimmune disease that does not require immunosuppressive medication and that, in the judgement of the site investigator (or designee), is likely not subject to exacerbation and likely not to complicate adverse event (AE) assessments. 19. Use of topical steroids over a total area exceeding 15 cm-2 within 30 days prior to Screening. 20. Treatment for an active AIDS-defining opportunistic infection within 90 days prior to Screening. 21. History of malignancy within the last 5 years. Note: A history of non-melanoma skin cancer (e.g., basal cell carcinoma or squamous cell skin cancer) is not exclusionary with documentation of complete resection at least 3 months prior to enrollment). 22. Compulsorily detained (involuntarily incarcerated) for treatment of either a psychiatric illness or a physical illness, e.g., infectious disease. 23. Known psychiatric, medical, occupational, or substance abuse disorders that would interfere with participant's ability to fully cooperate with the requirements of the trial as assessed by the study investigator (or designee). Specifically exclusionary: \[1\] recent psychosis; \[2\] ongoing risk for suicide; or \[3\] recent history of suicide attempt or gesture. 24. History or other clinical evidence of a significant medical condition that includes but is not limited to: a) A process that would affect the immune response b) A process that would require medication that affects the immune response c) Any contraindication to repeated injections, infusions, or blood draws d) A condition or process (e.g., chronic urticarial or recent injection or infusion with evidence of residual inflammation) for which signs and symptoms could be confused with reactions to the study product 25. Current anti-tuberculosis therapy 26. Diabetes Mellitus type 1 Note: Individuals with type 2 diabetes who meet inclusion criteria for glucose (Inclusion Criterion #23) are not excluded. 27. History of coronary artery disease, congestive heart failure, or cardiac arrhythmia requiring current treatment prior to study screening. 28. Hypertension - Exclude for blood pressure consistently \>150 mm Hg systolic and \>100 mm Hg diastolic. Note: Elevated BP occurring during research leukapheresis procedures completed within the past 12 months are excluded from this requirement. Acceptable isolated elevations must be noted as acceptable and signed by study PI or designee. 29. Unstable asthma (e.g., sudden acute attacks occurring without an obvious trigger) or either of the following in the past 12 months: 1. \>1 exacerbation of symptoms treated with oral/parental corticosteroids; 2. Emergency care, urgent care, hospitalization, or intubation for asthma. 30. Bleeding disorder diagnosed by a doctor (e.g., factor deficiency, coagulopathy, or platelet disorder requiring special precautions) 31. Seizure disorder: History of seizure(s) within past three years or use of medications used to prevent or treat seizure(s) at any time within the past 3 years. 32. History of asplenia - absence of normal spleen function as indicated by: 1. Splenectomy 2. Sickle cell disease 33. History of hereditary angioedema, acquired angioedema, or idiopathic angioedema. 34. Prisoner recruitment and participation is not permitted.