In patients with heart failure, the heart muscle does not pump well. Over a period of years, this continues to get worse until the patient dies of heart failure. The investigators are trying to find out if the injection of these BMPCs can make a change in the functioning of these areas of the heart muscle. Data from studies around the world have suggested that when patients with heart failure receive these cells by direct injection into their hearts they show signs of recovered heart function, however, there has been no evidence from actual studies of the cells of the patient's hearts to show how this process works. It is the investigator's plan to inject an eligible participant's heart with cells that are from one's own bone marrow during an operation to receive a ventricular assist device (VAD) or partial artificial heart and then to study the function of the heart while awaiting a heart transplant. The investigators will then examine the heart after it has been removed as part of the regular heart transplant operation for any microscopic changes (changes too small to be seen by the unaided eye but large enough to be studied under a microscope) at the site where the cells are injected. Participants will have no change in the chances of receiving a heart transplant by agreeing to participate in this study. There will also be no delay in receiving a VAD operation while waiting to participate in this study. This Phase I study has been cleared by the Food and Drug Administration (FDA) to enroll and treat patients. The Center for Biologics Evaluation and Research Investigational New Drug number (IND BB #) is 12304. (A Phase I trial is a research study using techniques or products in the first-stage or for the first time in human subjects).