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 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. These specialized cells may have the ability to promote blood vessel growth, prevent cell death, and transform themselves into a number of tissues, including muscle. After an acute heart attack, a remodeling process is initiated in the heart in an attempt to compensate for damaged areas. Consequently, the condition of the heart muscle several weeks after a heart attack may differ considerably from the heart's condition during the acute setting. For some patients, delaying the delivery of the stem cells until 2 to 3 weeks after a heart attack may be better than initiating treatment during the acute phase. This study will evaluate the safety and effectiveness of placing adult stem cells into injured heart muscle 2 to 3 weeks after a heart attack for improving heart function in people who have had a recent heart attack and a PCI.