Coronary artery disease (CAD), a disease in which blood vessels become clogged by a build-up of plaque, is the leading cause of heart failure, a condition in which the heart can no longer pump enough blood to the rest of the body. People with heart failure caused by CAD are said to have ischemic cardiomyopathy. Normal treatment for CAD involves coronary artery bypass grafting (in which a vein from another part of the body is grafted around an artery that has become blocked) or coronary angioplasty and stent placement (in which a blocked artery is opened and a small tube is placed to keep the artery open). However, some people with ischemic cardiomyopathy, such as those with substantial scar tissue on the heart wall or those with a particular heart structure, may not be eligible for these treatments. An alternative treatment being developed is therapeutic angiogenesis, which involves stimulating the growth of new blood vessels. Recent research has shown that withdrawing stem cells from bone marrow and implanting the cells into heart tissue may be an effective way to achieve therapeutic angiogenesis. This study will determine the safety and effectiveness of using stem cells to stimulate new blood vessel growth in the hearts of people with ischemic cardiomyopathy.