Adult Double Cord Blood Transplant Study

Inclusion Criteria: * Age 22 - 50 years * Patients will have one of the following hematological malignancies: * Acute myelogenous leukemia (AML): Complete first remission (CR1) at high risk for relapse or Complete second remission (CR2) * Acute lymphoblastic leukemia (ALL): Complete first remission (CR1) at high risk for relapse or Complete second remission (CR2) * Acute undifferentiated leukemia (AUL) or biphenotypic leukemia: CR1 or CR2 * Myelodysplastic Syndrome (MDS) with one of the following: Low and Intermediate-1 International Prognostic Scoring System (IPSS) score with Life-threatening neutropenia or thrombocytopenia; or Platelet transfusion dependence Intermediate-2 or High IPSS score Therapy-related disease: patient with history of chemotherapy and current evidence of MDS * Patients with adequate organ function and performance status criteria * Two Suitable Umbilical Cord Blood Units: a cryopreserved dose of at least 1.5 x 107 TNC/kg. If the unit contains red cells at time of cryopreservation, the cryopreserved dose must be at least 2.0 x 107 TNC/kg. Each unit must be at least 4/6 HLA-A and B antigen, and DRB1 allele matched with the recipient. Each unit must be at least 3/6 HLA-A, B DRB1 antigen matched to each other. Exclusion Criteria: * Patient with suitable related donor * AML, ALL, AUL, biphenotypic leukemia beyond CR2 * AML evolved from myelofibrosis * Any acute leukemia with: * Morphologic relapse or persistent disease in the BM * Active extra-medullary leukemia including active CNS leukemia * Requiring greater than two cycles of chemotherapy to obtain present remission status * Bone marrow aplasia (defined as BM cellularity \< 5% at transplant work-up) * MDS with 10% or greater bone marrow blasts at pre-transplant workup * Prior autologous or allogeneic HSC transplant at any time * Prior radiation therapy rendering patient ineligible for TBI * Any uncontrolled infection at time of study enrollment * Seropositive or NAT positive for HIV or HTLV1 * Females who are pregnant or breast feeding * Patient unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, follow-up, and research tests