Eligible participants are HIV positive and plan to be treated by high dose chemotherapy
        followed by an autologous stem cell transplant (ASCT). Participants are a minimum of 15
        years of age with Karnofsky performance status greater than or equal to 70% that have
        primary refractory or recurrent diffuse large B-cell, immunoblastic, plasmablastic, high
        grade, Burkitt, primary effusion lymphoma, or classical Hodgkin lymphoma. Participants must
        have received 2 or 3 prior treatment regimens, including an induction chemotherapy and 1 or
        2 salvage regimens. Monoclonal antibody therapy and local radiation will not be counted as
        prior therapies. Participants must have chemosensitive disease as demonstrated by complete
        or partial response to induction or most recent salvage chemotherapy. Participants cannot
        have had prior autologous, allogeneic HCT, or CART-cell therapy. Participants must initiate
        conditioning therapy within 3 months of stem cell mobilization or bone marrow harvest.
        Blood cell mobilization or bone marrow harvest will be carried out per institutional
        guidelines. Participants may not have HIV refractory to pharmacologic therapy. Patients
        must not have an uncontrolled infection. Participants must not have received previous
        cellular therapy

        Inclusion Criteria:

          1. Age 15 years old or older at time of enrollment.

          2. Receiving antiretroviral therapies (ART) with HIV viral load below the limit of
             detection by standard commercial assay. A single plasma HIV-1 RNA measurement that is
             ≥ the limit of quantification of the FDA-approved commercial assay but

        Exclusion Criteria:

          1. Karnofsky performance score less than 70%.

          2. Participant is known to have an HIV subtype other than B.

          3. Participant has documented raltegravir or protease inhibitor resistance.

          4. Myocardial infarction within 6 months prior to enrollment or New York Heart
             Association (NYHA) Class III or IV heart failure, uncontrolled angina, severe
             uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute
             ischemia.

          5. Uncontrolled bacterial, viral or fungal infection (currently taking medication and
             with progression or no clinical improvement).

          6. Participant has active CNS involvement.

          7. Participants with prior malignancies except resected non-melanoma skin cancer or
             treated cervical carcinoma in situ. Cancer treated with curative intent greater than
             or equal to 5 years previously will be allowed. Cancer treated with curative intent
             less than 5 years BMT CLINICAL TRIALS NETWORK HIV T-Cell - Protocol 1903 Version 1.0
             Dated February 24, 2021 2-4 Confidential previously may be eligible must be reviewed
             and approved by the Protocol Officer or Chairs.

          8. Female participants that are pregnant as per institutional definition or
             breastfeeding.

          9. Fertile men or women unwilling to use contraceptive techniques from the time of
             initiation of mobilization until six-months post-transplant.

         10. Prior autologous or allogeneic HCT, or prior therapy with chimeric antigen receptor
             (CAR) T-cells.

         11. Participants with evidence of MDS/AML or abnormal cytogenetic analysis indicative of
             MDS on the pre-transplant bone marrow examination. Pathology report documentation need
             not be submitted.

         12. Steroids greater than 0.5 mg/kg/day prednisone equivalents.

         13. Bone marrow involvement by lymphoma at time of workup. Prior history of bone marrow
             involvement is allowed if cleared prior to ASCT.