Recipients of a first or second cadaver kidney transplant are randomized pre-transplant to 1 of 2 treatment groups. One group receives antibody therapy (Thymoglobulin) at the time of transplant and delayed cyclosporine therapy. The other group starts cyclosporine therapy at the time of transplant without Thymoglobulin. DGF is diagnosed by a less than 20 percent decrease in the serum creatinine levels in the first 24 hours post-transplant and/or the need for dialysis. Patients on the antibody arm receive additional antibody if they experience DGF. Biopsies are performed in all cases of suspected rejection and any patient with biopsy-confirmed acute cellular rejection receives treatment. Patients have regular examinations including blood tests and are evaluated for kidney function and incidence of complications for 24 months after the transplant. The trial endpoint of graft function encompasses graft survival and graft function as calculated by creatinine clearance.