All individuals who receive a heart transplant are at risk for developing antibody-mediated rejection (AMR). An antibody is a protein produced by the body's immune system when it detects a foreign substance, called an antigen. The mechanism of an antibody is to attack an antigen. In antibody mediated rejection, antibodies will attack the transplanted heart, causing injury to the heart. The purpose of this investigation is to determine if a study drug, called Eculizumab (Soliris), prevents antibody-mediated rejection in patients with high antibody production and prolongs long-term cardiac transplant survival.