This study looks at the effects of two medications, alendronate and parathyroid hormone, on bone mass and on bone formation and bone breakdown in women with osteoporosis. We will randomly select postmenopausal women who have osteoporosis to receive laboratory-produced human parathyroid hormone (hPTH), or alendronate, or both for 2.5 years. Study participants will return to the study center periodically to have their bone mass measured and to give blood and urine samples for tests of bone formation and breakdown and for other laboratory tests. Those who complete the study are eligible for one or two 12 month extension studies.