Umbilical Cord Blood (UCB) Transplant, Fludarabine, Melphalan, and Anti-thymocyte Globulin (ATG) in Treating Patients With Hematologic Cancer

RATIONALE: Giving low doses of chemotherapy before a donor umbilical cord blood transplant helps stop the growth of cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. The donated stem cells may replace the patient's immune cells and help destroy any remaining cancer cells (graft-versus-tumor effect). Sometimes the transplanted cells from a donor can also make an immune response against the body's normal cells. Giving tacrolimus and mycophenolate mofetil after the transplant may stop this from happening. PURPOSE: This phase II trial is studying how well giving umbilical cord blood transplant together with fludarabine, melphalan, and antithymocyte globulin works in treating patients with hematologic cancer.