Type 1 diabetes, also known as insulin-dependent diabetes, is a chronic disease in which the pancreas produces insufficient insulin to properly regulate blood sugar levels. Hypoglycemia, low blood sugar, and hyperglycemia, high blood sugar, can lead to significant complications in people with type 1 diabetes. Intensive insulin therapy has been shown to reduce the risk of chronic complications in people who achieve near normalization of glycemia. However, this therapy is labor intensive, difficult to implement, and associated with an increased frequency of severe hypoglycemia. Transplantation of islets from a healthy pancreas has been successful in restoring normal blood sugar levels and has led to initial insulin independence in people with type 1 diabetes. Rejection of these islets by the recipient's immune system, however, makes the treatment ineffective within a couple of years. Immunosuppressant drugs may be an effective way to maintain islet function post-transplant. The purpose of this study is to assess the safety and efficacy of an immunosuppressive regimen that includes DSG on post-transplant islet function in people with type 1 diabetes who have not responded to intensive insulin therapy. The study will also seek to improve the understanding of determinants of success and failure of islet transplants for type 1 diabetes.