The goal of this phase I/II study is to investigate the safety and efficacy of a novel approach to allogeneic stem cell transplantation aimed at decreasing transplant related morbidity and mortality. We will treat patients with CGD with an allogeneic, G-CSF mobilized peripheral blood stem cell transplant from an HLA identical family member. The graft will be T-cell depleted which will decrease the incidence of acute graft vs. host disease. Donor T-cells will be infused post-transplant if engraftment of donor stem cells is unsatisfactory. The preparative regimen utilized will provide intense immunosuppression without myeloablation. It is designed to allow donor stem cell engraftment while minimizing serious transplant related toxicity.