Transplantation of donated stem cells (cells produced by the bone marrow that mature into the different blood components-white cells, red cells and platelets) can cure CGD. However, this procedure carries a significant risk of death, particularly in patients with active infection, because it requires completely suppressing the immune system with high-dose chemotherapy and radiation. In addition, lymphocytes-another type of infection-fighting white blood cell-from the donor may cause what is called graft vs. host disease (GvHD), in which the donor cells "see" patient's cells as "foreign" and mount an immune response to reject them. To try to reduce these risks, patients in this study will be given low-dose chemotherapy and no radiation, a regimen that is easier for the body to tolerate and involves a shorter period of complete immune suppression. Also, the donor's lymphocytes will be removed from the rest of the stem cells to be transplanted, reducing the risk of GvHD.