This study will investigate the safety and effectiveness of a modified stem cell transplant procedure for treating chronic granulomatous disease (CGD) in patients with active infection. CGD is an inherited disorder of neutrophils-a type of infection-fighting white blood cell-that leaves patients vulnerable to life-threatening infections. 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 recognize the patient's cells as foreign and mount an immune response to destroy 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.