Gene Therapy in Treating Patients With Human Immunodeficiency Virus-Related Lymphoma Receiving Stem Cell Transplant

Inclusion Criteria * Inclusion criteria associated with type and status of lymphoma, one of the following must be applicable: * Biopsy-proven intermediate or high-grade non-Hodgkin's lymphoma, meeting one of the following criteria (timeline 8 months prior to enrollment in the screening segment): \- In partial remission, \- Relapsed after initial complete remission, \- Failed induction therapy, but responds to salvage therapy (i.e., chemosensitive disease), \- In complete remission with high-risk features as specified by the International Prognostic Index. * Biopsy-proven advanced stage follicular lymphoma, that have failed at least two lines of therapy multi-agent chemotherapy, but responds to salvage therapy i.e.,chemosensitive disease) (timeline 8 months prior to enrollment in the screening segment). * Biopsy-proven advanced stage Mantle cell lymphoma with Ki-67 \> 10% in first complete remission (timeline 8 months prior to enrollment in the screening segment). * Biopsy-proven Hodgkin's lymphoma, meeting one of the following criteria (timeline 8 months prior to enrollment). \- In first, or greater relapse after initial complete remission, \- In partial remission, \- Failed induction therapy, but responds to salvage therapy (i.e., chemosensitive disease). * Biopsy-proven Burkitt's lymphoma, meeting one of the following criteria (timeline 8 months prior to enrollment): \- In second complete remission after relapse following initial complete remission, \- Failed induction therapy, but responds (very good partial remission, complete remission, or near complete remission) to salvage therapy (i.e., chemosensitive disease). * Biopsy proven plasmablastic lymphomas, or peripheral T cell lymphoma (with the exception of ALK+ type in first or second complete remission) \*NOTE: Patients meeting the following criteria are exempt from the 8-month timeline and do not require additional biopsy: * Patients who have never achieved a complete remission on the last biopsy-proven site of disease and went on to the next therapy then achieved a complete remission. * Patients who relapsed quickly (within 3 months of their last chemotherapy) and now have achieved a complete remission with salvage therapy. * Inclusion criteria associated with HIV-1 status * HIV-1 infection, as documented by any federally approved, licensed HIV rapid test performed in conjunction with screening (or ELISA, test kit, and confirmed by Western blot or other approved test). Alternatively, this documentation may include a record demonstrating that another physician has documented the participant's HIV status based on either: 1) approved diagnostic tests, or 2) the referring physician's written record that HIV infection was documented, with supporting information on the participant's relevant medical history and/or current management of HIV infection. * Must be on a multi-drug anti-HIV regimen (excluding zidovudine \[AZT, ZDV, Retrovir®, or agents containing zidovudine (e.g., Combivir® and Trizivir®)\], and efavirenz \[Sustiva®, or agents containing efavirenz (e.g., Atripla®)\]). Participants on zidovudine \[AZT, ZDV, Retrovir®; including Combivir® and Trizivir®\] and efavirenz \[Sustiva®; including Atripla®\] must switch to an alternative regimen without anticipated drug-drug interactions or myelosuppressive properties based on known viral resistance patterns and/or ART history, such as raltegravir and Truvada (emtricitabine and tenofovir) at least two weeks prior to the transplant. o Participant taking ARTs must satisfy one of the following: * Undetectable HIV viral load (\< 50 copies/mL). For patients who have had negative viral loads in the past 6 months and no known HIV viral load \>500 copies/mL within the last 6 months, minor fluctuations of viral load (isolated escalations up to 500 copies/mL) are acceptable. The participant's history of negative viral loads may be documented with recent laboratory results and/or a record from the participant's HIV care provider. * If viral load is detectable at \< 2000 copies/mL a review of previous antiretroviral regimens or previous genotypic or phenotypic testing which indicate the ability to fully suppress virus by addition of sensitive drugs must be performed. This review will be carried out by the protocol ID team or the ID specialist caring for the patient. * If viral load is detectable at ≥ 2000 copies/mL, a current HIV genotype and/or phenotype must be obtained. If a HAART regimen to which the patient's virus is sensitive can be determined based on genotype and previous antiretroviral experience, then the patient will be considered eligible in this regard. This review will be carried out by the protocol ID team or the ID specialist caring for the patient. General Inclusion Criteria (timeline: within 8 weeks prior to enrollment in the screening segment, unless otherwise specified) * Karnofsky performance status of 70-100%. ECOG performance status \<2 * SGOT and SGPT ≤ 2.5 times upper limit of normal (ULN). Serum bilirubin ≤ 2.5 times ULN except for participants who are on atazanavir or indinavir, or with elevated indirect bilirubin related to bilirubin conjugation issues such as Gilbert's disease, provided that the participant's direct bilirubin is within normal institutional limits. * Participants who are hepatitis C virus antibody positive, or hepatitis B virus surface antigen positive must be free of clinical evidence of cirrhosis as determined by the principal investigator in consultation with the institutional Gastroenterology Service. * Participants with Hepatitis B should be on appropriate anti-viral therapy at the time of the transplant, and their viral load should be negative. * Serum creatinine ≤ 2 times ULN. * Creatinine clearance ≥ 60 mL/min by the modified Cockcroft-Gault Formula. * PT/PTT ≤ 2 times upper limit of normal (ULN), or international normalized ratio (INR)/PTT ≤ 2 times the ULN. * FEV-1 or DLCO (corrected for hemoglobin) ≥ 50% predicted. * LVEF ≥ 50% by 2D ECHO or MUGA scan. * Not pregnant or nursing, with negative serum pregnancy test. Pregnant women are excluded from this study because the conditioning regimen has the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with BEAM, breastfeeding should be discontinued. These potential risks may also apply to other agents used in this study. * Participants should agree to practice effective contraceptive precautions and to use at least one method of contraception for the duration of the study and for 3 months post-transplant. * Age ≥18 years. Because only adult transplant centers are participating as study sites. * Life expectancy of greater than 3 months. * Ability to understand and the willingness to sign a written informed consent document. * Receipt of a stable ART regimen for at least 3 weeks prior to enrollment. Exclusion Criteria Participants who do not fulfill the criteria as listed above, are ineligible. Additionally, the presence of any of the following conditions will exclude a participant from study enrollment (timeline for all the exclusion criteria is within 8 weeks prior to enrollment in the screening segment): * Participants with \> 5% involvement of bone marrow by malignant cells (either by manual count or flow cytometry) prior to stem cell collection. * Participants with any abnormal cytogenetics in the bone marrow not related to the lymphoma, and not deemed to be constitutional. * Participants with unexplained anemia and/or thrombocytopenia. * Participants with clear evidence of myeloproliferative disorders, or myelodysplastic disorders in the marrow. * Presence of any active CNS disease at the time of evaluation (parenchymal or leptomeningeal). * Any history of HIV-1 associated encephalopathy. * Participants with persistently low CD4 counts less than 200 and a history of any AIDS-defining infection in the last 6 months before screening are excluded from the study. * Symptomatic/active bacterial, or fungal, or any other opportunistic infection. * Active CMV retinitis, or other active CMV-related organ dysfunction. * Relapse of pneumocystis carinii pneumonia within the past year before enrollment. * Intractable, severe diarrhea, defined as \> 1.500 cc diarrheal fluid per day, or diarrhea causing persistent severe electrolyte abnormalities, or hypoalbuminemia. * History of active myocardial ischemia, cardiomyopathy, uncontrolled dysrhythmia, or congestive heart failure within the last 6 months before enrollment. * Dementia of any kind. * Seizures within the past 12 months before enrollment. * History of Grade III hemorrhagic cystitis due to prior cyclophosphamide chemotherapy. * History of other prior malignancy, except squamous cell carcinoma of the cervix or anus, superficial basal cell or squamous cell skin cancer, or other malignancy curatively treated more than 5 years ago before enrollment. * Active psychosocial condition that would hinder study compliance and follow-up. * Any perceived inability to directly (and without the means of a legal guardian) provide informed consent. * Any medical or physical contraindication, or other inability to undergo HPC collection. * Participants who are receiving any other investigational agents. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements.