This study will test whether a new islet transplant procedure will enable patients with type 1 diabetes mellitus to stop insulin therapy. Islets are cell clusters in the pancreas that contain insulin-producing cells. The new procedure features three important advances, first developed by a group in Edmonton, Canada, over the way islet transplants have traditionally been performed: 1) the islets are transplanted immediately after they are removed from the donor; 2) islets are transplanted from two different donors in order to obtain the number of islets in a normal pancreas; and 3) the anti-rejection drug regimen is designed to reduce the harmful side effects of "conditioning" chemotherapy. (In the standard transplant procedure, patients receive intensive chemotherapy following the transplant. This study will use no radiation and lower-dose chemotherapy.)