Kidney transplantation is the best treatment for many patients with kidney failure. Sometimes a transplanted kidney is rejected by the patient's immune system. Many types of immune system cells, including B cells, are active in rejection. B cells produce antibodies against anything the body sees as non-self, like germs or a transplanted kidney. Most medicines that help prevent transplant rejection affect cells other than B cells. Belimumab is a medication used to treat a disease called lupus. Belimumab slows development of antibody-producing B cells. This study will test whether belimumab works on parts of the immune system that cause rejection. Twenty to thirty adults getting a kidney transplant will be in this study. Like flipping a coin, a computer will randomly assign half to be given belimumab and half to be given placebo (a fake medicine). Patients and doctors will not know which medicine was assigned until the study is over. A total of 7 doses of study medicine will be given through a vein. One dose will be given during transplant surgery, and the other 6 will be given 2, 4, 8, 12, 16 and 20 weeks after transplant surgery. Usual transplant medicines will also be given. After all of the doses have been given, patients will be watched and tested at 24, 36, and 52 weeks after the transplant surgery. Blood samples will be tested to see what study medicines do to the immune system in transplant patients. If patients get a kidney biopsy, the samples will be tested to see if belimumab had any effect. Patients will be asked many questions to see if they are having any side effects. The study will be done at Addenbrooke's Hospital in Cambridge and Guys &St Thomas Hospital in London, United Kingdom. A pharmaceutical company, GlaxoSmithKline, is funding the study.