Allogeneic bone marrow transplantation is an accepted therapeutic option for many hematologic, immunologic, and malignant diseases, including chronic myelocytic leukemia and acute leukemia during or after first relapse (second remission). In order to maximize the chance for a successful transplant, it is desirable that the donor and the recipient share the same Human Leukocyte Antigen (HLA) histocompatibility antigens. Because of the Mendelian inheritance of HLA antigens, the chances of finding a match are much greater among relatives than in the general population. However, only about 30 percent of patients needing a transplant have a matched sibling. Thus a transplant from an HLA-matched unrelated donor may be an important alternative for these patients.