Patients receive 9 fractions of total body irradiation (TBI) on days -9 to -5 followed by melphalan IV for three days on days -4 to -2 and antithymocyte globulin IV or methylprednisolone IV for three days on days -3 to -1. On day 0, patients receive umbilical cord blood infusion. If TBI is not tolerated, busulfan is substituted and administered orally every 6 hours for 4 days on days -8 to -5. Cyclosporine and methylprednisolone begin on day -2 and continue for 6 months.