 This study involves the use of a drug called Thymoglobulin, which is approved in the US to treat kidney transplant rejection and in Canada to treat and prevent kidney transplant rejection. This study will evaluate the effect of Thymoglobulin induction therapy and reduced doses of calcineurin inhibitors on the incidence of liver rejection and will provide a basis for future evaluations of Thymoglobulin as an immunosuppressive agent to help decrease the incidence of liver transplant rejection. Subjects meeting all inclusion and exclusion criteria are eligible to participate in this study. Approximately 75 study subjects from up to 18 transplant centers in the United States and Canada will be enrolled in this 12-month study. 