RATIONALE: Giving low doses of chemotherapy and total-body irradiation before a donor umbilical cord blood stem cell transplant helps stop the growth of cancer or abnormal cells. It also stops 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 or abnormal 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 before the transplant may stop this from happening. PURPOSE: This clinical trial is studying how well umbilical cord blood stem cell transplant works in treating patients with hematologic cancer or other disease.
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