Gene-Modified HIV-Protected Stem Cell Transplant in Treating Patients With HIV-Associated Lymphoma

Inclusion Criteria: * HIV-1 seropositive * Stable, continuous antiretroviral treatment for at least three months before enrollment, defined as a multi-drug regimen (excluding azidothymidine \[AZT\]) * HIV plasma viral load \< 50 copies/ml * Non-Hodgkin or Hodgkin lymphoma without active central nervous system (CNS) involvement associated with poor prognosis with medical therapy alone or for which autologous peripheral blood stem cell (PBSC) transplant is indicated: * Hodgkin's lymphoma beyond first remission or first partial remission or induction failure with subsequent response to salvage therapy * Non-Hodgkin's lymphoma beyond first remission or first partial remission or induction failure with subsequent response to salvage therapy * Chemotherapy responsive disease * Karnofsky performance score \>= 70% * Subjects must agree to use effective contraception from enrollment through completion of the study * Female subjects: if of child bearing potential, must have negative serum or urine pregnancy test within 7 days of enrollment * Subjects must be on a prophylactic regimen for pneumocystis carinii pneumonia, or agree to begin such treatment, if CD4+ cell counts are observed to be =\< 200/ul in peripheral blood * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Renal disease: serum creatinine \> 2 times upper limit of normal * Liver disease: alanine aminotransferase (ALT) or aspartate aminotransferase (AST) greater than 3 times the upper limits of normal, unless determined to be a result of the primary hematologic malignancy * Serum bilirubin greater than 3 times the upper limit of normal, unless attributed to Gilbert's syndrome * Pulmonary disease: forced vital capacity (FVC) \< 60% predicted * Pulmonary disease: forced expiratory volume in 1 second (FEV1) \< 60% predicted * Pulmonary disease: diffusion capacity of the lung for carbon monoxide (DLCO) parameters \< 60% predicted (corrected for hemoglobin) * Cardiac insufficiency: left ventricular ejection fraction (LVEF) \< 50% or coronary artery disease requiring treatment * Active infection requiring systemic antibiotic therapy with antibacterial, antifungal, or antiviral agents (excluding HIV) * Hepatitis B surface antigen positive * Hepatitis C virus (HCV) antibody positive and detectable HCV quantitative ribonucleic acid (RNA) with clinical evidence of cirrhosis * Requiring active treatment for Toxoplasma gondii * Planned radiation therapy after transplant * Malignancy other than lymphoma, unless (1) in complete remission and more than 5 years from last treatment, or (2) cervical/anal squamous cell carcinoma in situ or (3) superficial basal cell and squamous cell cancers of the skin * History of HIV-associated encephalopathy; dementia of any kind; seizures in the past 12 months; any perceived inability to directly provide informed consent (Note: Consent may not be obtained by means of a legal guardian) * A medical history of noncompliance with HAART or medical therapy * Any concurrent or past medical condition that, in the opinion of the Investigator, would exclude the subject from participation or any psychosocial conditions that would hinder study compliance or follow-up, at the discretion of the Investigator * Receipt of a vaccine for HIV-1 or any prior gene modified cell product, at any time * Known hypersensitivity to any of the products used in the trial-G-CSF (Neupogen), plerixafor, or any planned components of conditioning regimens * Pregnant or nursing women