X-linked severe combined immunodeficiency (SCID-X1) is a genetic disorder caused by defects in the common cytokine receptor chain, normally on the surface of lymphocytes. Individuals with SCID-X1 lack the normal development of a functional immune system and so have difficulty fighting infections, which may lead to chronic or severe illness and death. X-SCID patients are normally rescued by a bone marrow transplant from a healthy donor. This trial aims to treat SCID-X1 using a self-inactivating lentiviral vector carrying a functional gene to correct the genetic defect. By collecting an individual's stem cells and modifying them with a lentivirus, the gene-corrected cells can be returned into the blood to help produce normal healthy immune cells.