Warfarin has proven effective in patients with ischemic heart disease, especially in the reduction of stroke, death and re-infarction following myocardial infarction, and in the reduction of stroke in atrial fibrillation. Warfarin is the most promising unstudied intervention in patients with cardiac failure. This randomized, double-blind, multi-center study will define optimal antithrombotic therapy for patients with cardiac (heart) failure and patients with low ejection fraction (EF). EF is the proportion of left ventricular volume emptied during systole. It reliably measures left ventricular systolic function.