This study will last for 1 year after the final islet transplant. Participants may receive up to 3 islet transplants while participating in this study. Participants eligible for this study will have clinic visits every 6 months. Once a preparation of islets becomes available, participants will be randomly assigned to Arm 1 or Arm 2. Participants in Arm 1 will receive LMW-DS during and for 5 hours after infusion. Participants in Arm 2 will receive heparin at the time of infusion. In addition, all participants will receive anticoagulation prophylaxis agents consisting of Klexzane® (Enoxaparinsodium) and Trombyl® or Albyl-E® (Acetylsalicylic acid). All participants will also receive the oral immunosuppression medications consisting of mycophenolate mofetil or sirolimus and tacrolimus or cyclosporine throughout the study. In addition, they will receive intravenous thymoglobulin on days -2, -1, day 0 (transplant), +1, and +2 for the first transplant or intravenous basiliximab at the time of transplant and on Day 4 for the second and third transplant. Enbrel® (Etanercept) will be given to all participants for anti-inflammatory therapy. Islet infusions will occur at the hospital and will be given intravenously. Participants will be eligible to receive second and third islet infusions if previous infusions fail and they continue to meet the eligibility criteria. After each infusion, study visits will occur on Days 1, 3, 7, 14, 21, 28, and 75 and Months 6 and 12. At these visits, physical exams and blood collection will occur. At some visits, urine collection will also occur.