FA is a rare, inherited disease that is caused by a gene defect and that primarily affects an individual's bone marrow, resulting in decreased production of blood cells. The lack of white blood cells affects an individual's ability to fight infections, the lack of platelets may result in bleeding, and the lack of red blood cells usually leads to anemia. FA is typically diagnosed in childhood, and there is a high fatality rate. Bone marrow transplants are one common treatment for FA. However, there are many risks associated with transplantation, including rejection of the transplanted cells and graft-versus-host disease, a serious side effect in which donor cells attack the recipient's tissues. This study will use an experimental gene transfer procedure performed in a laboratory to insert a new FA gene into the participant's bone marrow cells. The gene-corrected bone marrow cells will then be re-infused into the participant and participants will be observed for successful gene transfer. The purpose of this study is to evaluate the safety and effectiveness of the FA gene transfer procedure and to determine the ability of the gene-corrected cells at generating new, healthy blood cells in individuals with FA.