The primary specfic aim of this randomized, double-blind, Multi-center clinical trial is to determine whether Warfarin (International Normalized Ratio [INR] 2.5-3.0, target INR 2.75) or Aspirin (325 mg per day) is superior for preventing all-cause mortality and stroke combined in patients with Ejection Fraction (EF) great than or equal to 30%, when balanced against any risk of Intracerebral Hemorrhage. The secondary aim is to answer the following questions:

(1) Is Warfarin or Aspirinsuperior for reducing stroke alone?

(2) Among women, is Warfarin or Aspirin therapy superior for reducing all-cause mortality, stroke, and MI combined, when balanced against any increased risk of intracerebral hemorrhage?

(3) Among African-Americans, is Warfarin or Aspirin therapy superior for reducing all-cause mortality, stroke, and MI combined, when balanced against any increased risk of intracerebral hemorrhage?

(4) Does any relative risk of benefit of Warfarin or Aspirin for stroke, TIA, or periheral embolism depend on Ejection fraction?

(5) Does any relative risk or benefit of Warfarin or Aspirin depend on the Heart failure class (NYHA class)?

(6) Does any relative risk or benefit of Warfarin or Aspirin depend on etiology of Cardiac failure?

(7) Do Warfarin and Aspirin differ in their effect on cognitive function?

(8) Does Warfarin have a larger risk reduction relative to Aspirin in cardioembolic than other stroke subtyped?

(9) Are cardioembolic infarcts more frequent in patients with non-ischemic than ischemic caridiac diseases?

(10) Are cardioembolic infarcts more frequent in women than in men?

(11) Is mean cerebral infarct volume greater in patients with EF less than or equal to 20% than in patients with EF greater than 20%?

A tertiary aim is to combine our data with that of the Warfarin Antiplatelet Trial in Chronic Heart failure (WATCH) to provide more statistical power to address the secondary aims 1 through 6.