Transplantation of donated stem cells (cells produced by the bone marrow that mature into the different blood components-white cells, red cells and platelets) is a very effective treatment for CML. However, despite its success in a large number of patients, there is still a significant risk of death from the procedure. In addition, it results in sterility and leaves patients at increased risk for other cancers and for eye cataracts. These complications result from the intensive chemotherapy and radiation patients receive before the transplant to rid the body of cancer cells. In this study, radiation will not be used and chemotherapy drugs will be given in lower doses to try to reduce the dangers of the procedure.