Participation in this study will last 24 months. All participants will first undergo baseline assessments that will include a medical history, a physical exam, an electrocardiogram (ECG), blood draws, an echocardiogram, and a magnetic resonance imaging (MRI) test. Participants will then be assigned randomly to receive stem cells or placebo between 2 and 3 weeks after their heart attack. The morning of the stem cell or placebo infusion, participants will undergo a blood draw and a bone marrow aspiration procedure of the hip bone to collect the stem cells. Later the same day, either stem cells or placebo will be infused through a catheter and into the damaged area of the heart.