Previous studies have shown that topiramate is effective in preventing migraine headaches. This study will start with a 4-week baseline period in which no treatment is given, followed by a 26-week period in which each patient's dose is adjusted and then kept stable for 4 weeks. The dose will start at 25 milligrams of topiramate per day and will be increased 25 milligrams per day once weekly and then raised to either the target--100 milligrams per day--or the maximum dose that is well tolerated up to 200 milligrams per day. Patients randomized to receive topiramate will remain on that dose. The comparison phase of the study is a 26-week period in which the change in migraine days of patients on topiramate (taking at least 50 milligrams per day) is compared with the change in migraine days for patients on the placebo. Also studied will be the patients' health-related quality of life as assessed by questionnaires filled out at specific visits as well as the patients' views of the safety and tolerability of topiramate. The study hypothesis is that the number of migraine days, periods, and attacks during the last 4 weeks of the double-blind phase, relative to the last 4 weeks of the open-label phase, is reduced more in the topiramate group than the placebo group. During open-label (26-weeks) and double-blind phase (26-weeks), patients receiving topiramate will take 25 milligrams to 100 milligrams daily by mouth; increased by 25 milligrams per day once weekly; dose cannot exceed 200 milligrams per day and must be stable for the last 4 weeks.