Prevention of stroke in patients with narrowing of one of the arteries in the brain typically consists of using medications to prevent blood clots from forming. Currently, the best medication to use in this situation is unknown. The purpose of this study is to compare the effectiveness of two different medications, warfarin or aspirin, for the prevention of stroke due to narrowing of one of the large arteries in the brain. Patients must have experienced a recent transient ischemic attack (TIA) or mild stroke. Stroke of this type is thought to occur more often in minorities.