Following screening procedures and 2 days prior to islet transplant, participants will be randomly assigned to either this Phase 2 trial or a multicenter Phase 3 trial. Participants in this study will receive up to three separate islet transplants. They will begin receiving antithymocyte globulin (ATG) and sirolimus 2 days prior to the first islet transplant. ATG will continue to be given until Day 2 post-transplant. Participants will continue taking sirolimus for the duration of the study. On the day of transplant, participants will receive DSG and etanercept, in addition to ATG and sirolimus. The DSG infusion will be administered over 3 hours and will immediately precede the islet transplant. Participants will continue receiving daily 3-hour infusions of DSG through Day 6 post-transplant. Etanercept will also be administered on Days 3, 7, and 10 post-transplant. Tacrolimus will be administered on Day 1 post-transplant and continued throughout the study.