Brief Summary: This phase I trial evaluates the side effects and usefulness of axicabtagene clioleucel (a CAR-T therapy) and find out what effect, if any, it has on treating patients with HIV-associated aggressive B-cell non-Hodgkin lymphoma that has come back (relapsed) or not responded to treatment (refractory). T cells are infection fighting blood cells that can kill tumor cells. Axicabtagene ciloleucel consists of genetically modified T cells, modified to recognize CD-19, a protein on the surface of cancer cells. These CD-19-specific T cells may help the body's immune system identify and kill CD-19-positive B-cell non-Hodgkin lymphoma cells.

Detailed Description: PRIMARY OBJECTIVE: I. To demonstrate safety and feasibility of axicabtagene ciloleucel for relapsed/refractory (R/R) human immunodeficiency virus (HIV)-associated aggressive B-cell non-Hodgkin lymphoma (B-NHL) in participants with well-controlled HIV. SECONDARY OBJECTIVES: I. To estimate efficacy of axicabtagene ciloleucel in HIV-associated B-NHL as measured by complete response rate, event-free survival, and duration of response. II. To assess the relationship between T-cell subset profile and clinical response in R/R HIV-associated B-NHL treated with axicabtagene ciloleucel. EXPLORATORY OBJECTIVES: I. To assess the relationship between immune-mediated response and clinical response and toxicity in HIV-associated B-NHL treated with axicabtagene ciloleucel. II. To define burden of HIV integration in the final chimeric antigen receptor (CAR) T-cell product versus the pheresis product through quantitative polymerase chain reaction (PCR). OUTLINE: Patients receive fludarabine intravenously (IV) over 30 minutes and cyclophosphamide IV over 1 hour on days -5, -4, and -3. Patients then receive axicabtagene ciloleucel IV over 30 minutes on day 0. After completion of study treatment, patients are followed up once weekly for month 1, once a month for months 2-6, and then at 9, 12, 15, 18, and 24 months.