Type I 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, or low blood sugar, and hyperglycemia, or 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, can make the treatment ineffective. An immune response known as instant blood-mediated inflammatory reaction (IBMIR) results in the disruption of islet integrity and islet loss within an hour of transfusion. LMW-SD inhibits IBMIR by preventing the cascade that triggers it, when combined with pancreatic islets. The purpose of this study is to determine the safety and efficacy of LMW-SD given with islet transfusion and post-transfusion along with immunosuppressive therapy, including mycophenolate mofetil or sirolimus, tacrolimus or cyclosporine, and thymoglobulin or basiliximab, on the success of islet transplantation in people with type 1 diabetes.