In this clinical trial the researchers will try to answer this question. Women with hormone receptor positive breast cancer who become postmenopausal with chemotherapy will be invited to participate in this study. Each woman will be treated with one of the aromatase inhibitors, anastrozole (Arimidex), and then carefully monitored to ensure that her ovaries do not start making estrogen. If her estrogen level remains low, then she will continued to be followed for 18 months. If the level increases to the level typically seen in premenopausal women, however, then she will stop taking part in this study.