Heart attacks are a leading cause of death for both men and women in the United States. A heart attack occurs when blood flow to the heart is restricted, commonly due to a blood clot that has formed in one of the coronary arteries. If the clot becomes large enough, blood flow to the heart can be blocked almost completely and the heart muscle in that area can suffer permanent injury or death. Although a percutaneous coronary intervention (PCI) can be used to open up the blocked artery and restore blood flow to the heart muscle, there may be a significant amount of heart tissue that has been irreversibly damaged. Recent studies have shown that adult stem cells from bone marrow may be able to improve heart function after a heart attack. This study will evaluate the safety and effectiveness of using adult stem cells for improving heart function in people who have had a recent heart attack and a PCI.