The goal of this clinical trial is to study the effects of discontinuation of Alendronate treatment for three years. The participants are postmenopausal women with osteoporosis, who have had a treatment period without fractures. The study will answer: * What is the rate of fragility fractures during the study? The hypothesis is that discontinuation of Alendronate after a fracture-free period does not raise the 3-year fracture risk. * Are there predictors of significant bone loss after discontinuation of Alendronate? The hypothesis is that clinical, biochemical, cellular, or molecular characteristics can predict bone loss. Researchers will compare the outcomes of discontinuation of Alendronate with continuing treatment. Participants will be randomly assigned to a group. Participants will: * Either take alendronate once weekly for three years OR discontinue their treatment * Visit the clinic at 6 and 18 months for blood samples * Visit the clinic at the start of the study and after 1, 2, and 3 years for bone scans and blood samples.