Low-Dose Fludarabine, Busulfan, and Anti-Thymocyte Globulin Followed By Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer

RATIONALE: Giving chemotherapy before a donor umbilical cord blood transplant helps stop both the growth of cancer cells and the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient, they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells. Giving chemotherapy, such as fludarabine and busulfan, and antithymocyte globulin before transplant and tacrolimus and mycophenolate mofetil after transplant may stop this from happening. PURPOSE: This clinical trial is studying how well giving low-dose fludarabine and busulfan together with anti-thymocyte globulin, followed by donor umbilical cord blood transplant works in treating patients with advanced hematologic cancer.