Islet transplantation can provide physiologic insulin replacement to patients with type 1 diabetes without the complications associated with whole pancreas transplantation. The purpose of this study is to achieve insulin-independence in patients with type 1 diabetes, thereby eliminating the need for exogenous insulin injections to maintain normal glucose levels, ameliorating severe hypoglycemia and potentially decreasing the development of diabetes-related complications. This study will investigate islet transplantation in subjects who have preserved renal function and subjects who have undergone cadaveric renal transplantation, since the latter subjects are already on immunosuppression.