In order to prevent organ rejection, patients receiving liver transplants currently require life-long treatment with immune system-suppressing medications to prevent the rejection of the transplanted liver. However, these medications can cause long-term side effects, such as infection, kidney problems, diabetes, and cancer. In patients infected with hepatitis C virus (HCV), these medications may increase the risk of HCV infection in the transplanted liver. The purpose of this study is to determine whether a slow withdrawal of immune system-suppressing medications is safe in two groups of subjects: those who receive a liver transplant due to HCV, and those who receive a liver transplant due to non-immune, non-viral causes of liver failure. The study will also look at whether slow withdrawal will help reduce the long-term side effects of immune system-suppressing medications and decrease the chance for HCV infection of the new liver in transplant patients with HCV.