An investigator initiated, randomized, double-blind, multi-center, clinical trial designed to test the primary null hypothesis that warfarin therapy will not reduce the frequency of death or ischemic stroke recurrence by 30% compared with aspirin therapy. The trial will also test that the rate of death and recurrent stroke with warfarin will not be 50% lower in one year compared with aspirin therapy; there will be no differences between the two therapies in the frequency of recurrence by stroke severity or subtype, race or sex.