OUTLINE: Patients undergo a back-up bone marrow harvest prior to treatment. Patients receive 9 fractions of total body irradiation (TBI) on days -9 to -5, followed by melphalan IV on days -4 to -2 and antithymocyte globulin IV or methylprednisolone IV on days -3 to -1. If TBI is not allowed, busulfan is substituted and administered orally every 6 hours for 4 days on days -8 to -5. On day 0, patients receive umbilical cord blood infusion. Cyclosporine and methylprednisolone begin on day -2 and continue for 6 months. Patients are followed indefinitely for survival and late toxicity.