This study aims to determine if a new method can be used to treat Artemis-deficient Severe Combined Immunodeficiency (ART-SCID), a severe form of primary immunodeficiency caused by mutations in the DCLRE1C gene. This method involves transferring a normal copy of the DCLRE1C gene into stem cells of an affected patient. Participants will receive an infusion of stem cells transduced with a self-inactivating lentiviral vector that contains a normal copy of the DCLRE1C gene. Prior to the infusion they will receive sub-ablative, dose-targeted busulfan conditioning. The study will investigate if the procedure is safe, whether it can be done according to the methods described in the protocol, and whether the procedure will provide a normal immune system for the patient. A total of 25 patients will be enrolled at the University of California San Francisco in this single-site trial, and will be followed for 15 years post-infusion. It is hoped that this type of gene transfer may offer improved outcomes for ART-SCID patients who lack a brother or sister who can be used as a donor for stem cell transplantation or who have failed to develop a functioning immune system after a previous stem cell transplant.