The purpose of this study is to determine the effectiveness of teriparatide (FORTEO), which is human parathyroid hormone 1-34, for increasing bone mass and improving bone structure in adults affected with Osteogenesis Imperfecta (OI). Osteogenesis imperfecta is an inherited disorder of type I collagen, a major component of bones, and is characterized by multiple fractures and deformities. OI affects approximately 1-2 of every 10,000 individuals. Virtually all of the studies of potential treatments for OI have evaluated the effects of medications only on children with OI. There is no cure for osteogenesis imperfecta and there is no established medical therapy for adults with the disorder. There are very limited data concerning the usefulness of parathyroid hormone therapy in OI. An effective anabolic therapy for the treatment of adult patients with OI could be a valuable asset to the affected patients. In this study, the working hypothesis is that individuals affected with OI who are treated with Forteo will experience increased spine and hip bone mineral density and an increase in bone strength. Although Forteo is not expected to change the defect in the collagen produced, but is postulated to increase the quantity of bone formed and improve bone strength.