T-Regulatory Cell and CD3 Depleted Double Umbilical Cord Blood Transplantation in Hematologic Malignancies

Inclusion Criteria: * Only patients requiring a double umbilical cord blood (UCB) transplant are to be considered for this study. UCB Requirements * Three UCB units are required - one for Treg production and two for UCB transplant. The unrelated UCB donors must be 4-6/6 HLA-A, B, DRB1 matched with the recipient (HLA matching using molecular techniques: A and B to antigen level resolution and DR to allele level resolution). Suitable UCB units will be selected according to the University Of Minnesota UCB Graft Selection Algorithm. * Suitable UCB units must be ABO matched. Disease Criteria: * Patients aged 18 to 55 years * Acute Myeloid Leukemia: with morphologically persistent disease in a representative bone marrow aspirate sample with ≤ 10% blasts after at least 1 cycles of chemotherapy (if patient refuses or is disqualified from alternative protocols), or in 3rd or higher complete remission (CR). * Acute Lymphocytic Leukemia: with morphologically persistent disease in a representative bone marrow aspirate sample with ≤ 10% blasts after at least 1 cycles of chemotherapy, or in 3rd or higher CR * Chronic Myelogenous Leukemia in Blast Crisis: with ≤10% residual blasts in the bone marrow aspirate after at least 1 cycle of induction chemotherapy in combination with a tyrosine kinase inhibitor (TKI) * Refractory Anemia with Excess Blasts: (≤ 10%) in representative bone marrow aspirate sample of blasts after 1 cycle of induction chemotherapy. If treated with hypomethylating agents, patients are eligible if blast count is ≤ 10% after 4 cycles or evidence of stable or progressive disease after at least 2 cycles. * Chronic Myeloproliferative Disease * Chronic Lymphocytic Leukemia/Small Lymphocytic Lymphoma, Marginal Zone B-Cell Lymphoma or Follicular Lymphoma: disease must be refractory after at least two chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of ≥ 5 cm * Lymphoplasmacytic Lymphoma, Mantle-Cell Lymphoma, Prolymphocytic Leukemia: disease must be refractory after at least two chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of ≥ 5 cm * Large Cell Non-Hodgkin's Lymphoma: disease must be refractory after at least two chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of ≥ 5 cm * Lymphoblastic Lymphoma, Burkitt's Lymphoma, and other High-Grade NHL: disease must be refractory after at least two chemotherapy regimens or is chemotherapy sensitive but has residual nodal disease of ≥ 5 cm * Performance Status, Age, and Organ Function * Adequate performance status defined as a Karnofsky score ≥ 80% * Adequate organ function defined as: * Renal: creatinine \< 2.0 mg/dL, * Hepatic: bilirubin, AST/ALT, ALP \< 5 x upper limit of normal, * Pulmonary function: DLCOcorr \> 50% normal, * Cardiac: left ventricular ejection fraction \> 45% * Voluntary written informed consent signed before performance of any study-related procedure not part of normal medical care Exclusion Criteria: * Available medically suitable HLA-identical related donor * Active infection at time of transplantation (including active infection with Aspergillus or other mold within 30 days) * History of HIV infection * Pregnant or breast feeding. The agents used in this study may be teratogenic to a fetus and there is no information on the excretion of agents into breast milk. Females of childbearing potential must have a blood test or urine study within 14 days prior to registration to rule out pregnancy * Prior myeloablative transplant within the last 6 months * Extensive prior therapy including \> 12 months alkylator therapy or \> 6 months alkylator therapy with extensive radiation * Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tositumomab (Bexxar) as part of their salvage therapy (not eligible for myeloablative umbilical cord blood transplant)