Rheumatoid arthritis disease is believed to be due to immune cells, cells that normally protect the body and are now causing damage to the body. Risk of death is highest in people with twenty or more joints actively involved with disease, positive rheumatoid factor, an elevated sedimentation rate (laboratory measures of active inflammation), and patients with limitation of daily activities (trouble doing simple things like opening a carton of milk). In these high risk patients, life is significantly shortened. Death is usually from heart disease, kidney failure, neck dislocation, broken hip bones, or blood clots to the lung. In this study we use moderate dose chemotherapy (cyclophosphamide and fludarabine) and CAMPATH-1H (a protein that kills the immune cells that are thought to be causing the disease), followed by infusion of blood stem cells that have been collected from the patient's brother or sister (allogeneic stem cell transplant). The purpose of the moderate dose chemotherapy and CAMPATH-1H is to destroy the cells in the immune system and to allow the cells from the patient's brother or sister to grow. The purpose of the stem cell infusion is to restore blood cell production, which will be severely impaired by the moderate dose chemotherapy and CAMPATH-1H, and to produce a normal immune system that will no longer attack the body.