Twenty people will be evaluated in this pilot protocol. Ten will receive living donor kidney allografts and ten will receive cadaveric kidney allografts. Patients will be treated with Thymoglobulin beginning prior to graft implantation and continuing for ten days. Glucocorticosteroids will be given during the Thymoglobulin treatment to limit monocyte activation and prevent the cytokine release syndrome associated with this antibody preparation. Patients will be given Sirolimus orally beginning the day after transplantation and continuously thereafter. Patients will then be monitored for evidence of allograft rejection using standard functional parameters and protocol allograft biopsies. In addition, patients will be followed for a specific desired effect, allospecific AICD, that should promote the development of allospecific graft tolerance. This will be accomplished by assaying peripheral blood and allograft biopsies for apoptosis and peripheral blood for evidence of alloreactive T cell clone depletion.