Low blood sugar (hypoglycemia) is a recurrent problem for many people with diabetes. Successful transplantation of clusters (islets) of normal cells, that include those which produce the sugar-lowering hormone insulin, from the pancreas of a person who did not have diabetes into a person with diabetes should eliminate high blood sugar levels. We wish to determine if it will also eliminate low blood sugar. To do so we will give insulin to lower the blood sugar, measure the levels of the hormones that normally raise blood sugar levels (e.g., glucagon and epinephrine) and then stop the insulin and see if blood sugar levels return to normal. Because we anticipate that the transplanted islets will produce insulin, but not glucagon, this study may also tell us if regulated insulin production alone can prevent hypoglycemia in humans.