Cord Blood Transplant With Dilanubicel for the Treatment of HIV Positive Hematologic Cancers

Inclusion Criteria: * \>= 6 months to =\< 65 years * Treatment with combination antiretroviral therapy (cART) for at least 1 month before enrollment * Viral load \< 5000 copies/ml plasma on cART * Disease criteria * Acute myeloid leukemia * High risk in first complete remission (CR1), \>= 2 cycles to obtain complete remission (CR), erythroblastic or megakaryocytic leukemia; \>= in second complete remission (CR2) * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15% * Patients for whom adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible. Specimen for morphologic assessment, including possible repeat procedures will be obtained (as possible). These patients must be discussed with the lead principal investigator, Filippo Milano prior to enrollment * Acute lymphoblastic leukemia * High risk CR1 (for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed-lineage leukemia \[MLL\] rearrangements, hypodiploid); greater than 1 cycle to obtain CR; \>= CR2 * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15% * Patie…