At the present time, heart transplant recipients must take anti-rejection medication to prevent rejection of the donated heart. Even with this medication, chronic rejection is the most common cause of late graft loss. The anti-rejection agents themselves are significantly toxic, with side effects including kidney damage, infection and an increased incidence of cancer. The goal of this study is to allow the patient to develop "tolerance" to the transplanted heart while maintaining a competent immune system. Tolerance enables the transplant recipient's body to recognize the transplanted organ as self rather than foreign tissue. The recipient will not try to reject the donor heart and the need for anti-rejection medication could be dramatically decreased or eliminated entirely. To accomplish this, patients in this study will receive specially treated bone marrow taken from their heart donor. Bone marrow transplant has been shown in animal studies and in humans to induce tolerance following organ transplant.