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 4mg/kg/d x 1 beginning on day 12 and then 2.5 mg/kg/d for an additional 13 days. This trial expands on pilot studies at the NIH of 17 patients in which Campath was dosed both prior to and after transplantation with and without DSG. In those studies, excellent peripheral depletion occured after just one dose of Campath though central depletion required additional dosing. Thus, the goal of pre-reperfusion depletion can be achieved with a single pre-operative dose but thorough depletion requires additional post-operative dosing. Lasting rejection-free survival was not realized without the addition of some, albeit reduced immunosuppression. This is thought to be due to residual post-operative monocytes that infiltrated the allograft causing modest reversible allograft dysfunction. The current dosing regimen with DSG is thus designed to accomplish both pre-operative depletion, and more thorough post operative elimination of donor and recipient cells mobilizing as a result of reperfusion, combined with therapy aimed at preventing the activation of monocytes that escape depletion. The timing of the DSG is meant to correspond with the peripheral repopulation of monocytes seen in previous patients.