More than 1 million Americans suffer heart attacks each year. 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. 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. 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 using adult stem cell infusions 2 to 3 weeks after a heart attack for improving heart function in people who have had a recent heart attack and a common procedure called a percutaneous coronary intervention (PCI).