This phase II trial is studying the safety and potential efficacy of infusing non-human leukocyte antigen matched ex vivo expanded cord blood progenitors with one or two unmanipulated umbilical cord blood units for transplantation following conditioning with fludarabine phosphate, cyclophosphamide and total body irradiation, and immunosuppression with cyclosporine and mycophenolate mofetil for patients with hematologic malignancies. Chemotherapy, such as fludarabine phosphate and cyclophosphamide, and total-body irradiation given before an umbilical cord blood transplant stops the growth of leukemia cells and works to prevent the patient's immune system from rejecting the donor's stem cells. The healthy stem cells from the donor's umbilical cord blood help the patient's bone marrow make new red blood cells, white blood cells, and platelets. It may take several weeks for these new blood cells to grow. During that period of time, patients are at increased risk for bleeding and infection. Faster recovery of white blood cells may decrease the number and severity of infections. Studies have shown that counts recover more quickly when more cord blood cells are given with the transplant. We have developed a way of growing or "expanding" the number of cord blood cells in the lab so that there are more cells available for transplant. We are doing this study to find out whether or not giving these expanded cells along with one or two unexpanded cord blood units is safe and if use of expanded cells can decrease the time it takes for white blood cells to recover after transplant. We will study the time it takes for blood counts to recover, which of the two or three cord blood units makes up the patient's new blood system, and how quickly immune system cells return.