Transplantations will involve an inpatient hospital stay and infusion of islets into a branch of the portal vein. Participants who do not achieve or maintain insulin independence by Day 75 post-transplant will be considered for a second islet transplant. Participants who remain dependent on insulin for longer than 1 month after the second transplant and who show partial graft function will be considered for a third transplant. Daclizumab or basiliximab will be used in place of ATG for the second and third transplants, if they are necessary. Participants who do not meet the criteria for a subsequent transplant and do not have a functioning graft will enter a reduced follow-up period.