This trial aims to investigate the safety and efficacy of using gene therapy to treat subjects with β-Thalassemia Major. This trial being sponsored by Bluebird Bio and proposes to recruit 15 patients between July 2013 and July 2017. Participants in the study will receive their own blood stem cells (or CD34+ haematopoietic stem sells) that have been modified to include the proprietary product - LentiGlobin® BB305. The study seeks to explore if the product can result in the production of a protein that is absent in patients with β-Thalassemia Major, thus reversing this common condition. Participants will be monitored for complications following the transfusion and changes in clinical condition, including production of desired haemoglobin [β-A(T87Q)-globin] 18 to 24 months post treatment.