Congestive heart failure affects 5 million people in the United States, and over 60,000 are diagnosed with end-stage heart failure. Treatment options for these individuals are extremely limited. Because of a limited supply of donor organs, fewer than 3,000 people receive heart transplants each year. LVADs are battery-operated devices that can act as a temporary solution until a donor heart becomes available. Upon being surgically implanted into a weakened heart, the LVAD mechanically pumps blood from the heart to the rest of the body. There are, however, serious risks associated with LVADs, including infection, blood clots, and stroke. Early removal of the LVAD may be possible by injecting stem cells to regenerate new heart cells and improve heart function. The purpose of this study is to compare the safety and effectiveness of two types of bone marrow cells -- bone marrow mononuclear cells (BMCs) and immunoselected CD34+ hematopoietic stem cells -- in improving heart function in individuals with end-stage heart failure.