More than 1 million Americans suffer a heart attack each year, resulting in about a 38% mortality rate. Although current treatments are able to stabilize the condition of the heart, none is able to restore heart function as it was prior to the heart attack. The permanent damage to the heart can lead to more severe problems, such as heart failure and irregular heartbeat, making the discovery of treatments to improve heart function after a heart attack important. Adult stem cells, which are immature cells that can become many different types of cells, may offer a potential means of reversing or preventing permanent damage caused by 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. Recent studies have shown promise in using adult stem cells from bone marrow to reverse damage to the heart muscle caused by a heart attack, but more research is needed to assess the safety and effectiveness of stem cell use and to discover the best time to administer treatment. This study will evaluate the safety and effectiveness of placing adult stem cells into injured heart muscle for improving heart function in people who have had a recent heart attack and a PCI. Additionally, this study will help determine the best time to insert stem cells after a heart attack.