Recipients of living or cadaveric donor kidneys will be treated with one dose of Campath-1H prior to transplantation to insure that peripheral depletion is achieved at the time of graft reperfusion. Three subsequent doses of Campath-1H will be administered on the first, third and fifth days after the transplant to deplete passenger donor leukocytes and residual recipient cells that mobilize in response to the allograft. In addition, patients will be treated with DSG for 14 days beginning on the day prior to surgery. This trial expands on pilot studies at the NIH of 15 patients in which Campath was given alone at the time of transplantation. In those studies, excellent peripheral depletion occurred after just one dose of Campath though central depletion required additional dosing. This allowed for greatly reduced immunosuppression to be used to prevent rejection, but to date, all patients have required some immunosuppressive medication. It is hoped that the addition of DSG will eliminate the need for long-term immunosuppression.