Safety and Effectiveness of Low Molecular Weight Sulfated Dextran in Islet Transplantation After Kidney Transplant

Inclusion Criteria: * Subjects able to provide written informed consent and comply with study procedures * Clinical history compatible with T1D and onset of disease at less than 40 years of age and insulin dependence for more than (\>) 5 years at the time of enrollment; AND the sum of subject age and insulin-dependent diabetes duration is \>=28 * Absent stimulated C-peptide (less than 0.3 ng/mL) 60 and 90 minutes post-mixed-meal tolerance test (MMTT) * Subjects \>6 months post renal (kidney) transplant, currently taking or willing to switch to appropriate maintenance immunosuppression * Stable renal function and free of rejection for \>=3 months prior to islet transplantation * Standard medical treatment for \>=3 months under the care of an experienced diabetologist and at the end of this period has had at least 1 severe hypoglycemic event OR a hemoglobin A1C (HbA1c) \>7.2% OR reduced awareness of hypoglycemia manifest by a Clark score of \>=4 in the last year prior to study entry Exclusion Criteria: * Known immunoglobulin E (IgE) mediated allergy to antibiotics used in islet culture medium * Known hypersensitivity to dextran * Measured glomerular filtration rate (GFR) using Iothalmate, 51Cr-EDTA, 99-technetium-DPTA, or iohexol of less than 40ml/min/1.73 m\^2 * Proteinuria (albuminuria greater than 500 mg in 24 hours) of new onset since kidney transplantation * Other (non-kidney) organ transplants except prior failed pancreatic graft * Body mass index (BMI) \>30 kg/m\^2 * Insulin requirement of \>1.0 IU/kg/day * Consistently abnormal liver function tests (aspartate aminotransferase(AST), alanine aminotransferase (ALT),alkaline phosphatase, or total bilirubin) of greater than 1.5 times the upper limit of normal for two consecutive measurements that are \>2 weeks apart * Untreated proliferative diabetic retinopathy * History of hypercoagulability disorder or coagulopathy or International Normalized Ratio(INR) that is \>1.5 * Activated Protein C Resistance (APC-R) * Evidence by serologies and PCR of acute or chronic active Epstein-Barr Virus (EBV) infection OR no evidence by EBV serologies of prior EBV exposure * Any history of malignancy except for completely resected squamous or basal cell carcinoma of the skin * Known history of severe co-existing cardiac disease, characterized by any one of the following conditions: 1. Recent myocardial infarction (\<=6 months) 2. Evidence of ischemia on functional cardiac exam \<=last year 3. Left ventricular ejection fraction \<30% * Active infections, unless treatment is not judged necessary by the investigators(including but not limited to mild skin and nail fungal infections) * Active infection including hepatitis B, hepatitis C,human immunodeficiency virus (HIV), or pulmonary tuberculosis * Subjects with active peptic ulcer disease, symptomatic gallstones or portal hypertension * Acute or chronic pancreatitis * Subjects who are pregnant or breastfeeding, or who intend to become pregnant * Sexually-active females who are not: a) post-menopausal, b) surgically sterile, or c) using an acceptable method of contraception: oral contraceptives, Norplant, Depo-Provera, and barrier devices combined with spermicidal gel are acceptable; condoms used alone are not acceptable. * Active alcohol or substance abuse * Evidence of high-level sensitization (Panel Reactive Antibodies (PRA) \>50%) or positive cross match or the known presence of anti-donor HLA class I antibodies * Treatment with any anti-diabetic medication, other than insulin, \<=4 weeks of enrollment * Use of any investigational agents \<=4 weeks of enrollment * Receipt of live attenuated vaccine(s) \<=2 months of enrollment * Subjects with any condition or circumstance that in the opinion of the investigator would make it unsafe to undergo an islet transplantation