Complete DiGeorge anomaly is a congenital disorder characterized by athymia. Without successful treatment, children remain immunodeficient and usually die by age 2 years. In infants with complete DiGeorge anomaly and no T cells, thymus transplantation without immunosuppression resulted in diverse T cell development and good T cell function. Some infants with no thymus have some T cells that presumably developed extrathymically; these T cells can reject a thymus graft. The purpose of this study is to design better immunosuppression use for complete DiGeorge anomaly subjects who have some T cells and different T cell function levels. This protocol includes 3 immunosuppression regimens to allow subjects with different T cell function levels to be suppressed adequately.