A conduit is provided to provide a bypass around a blockage in the coronary artery. The conduit is adapted to be positioned in the myocardium or heart wall to provide a passage for blood to flow between a chamber of the heart such as the left ventricle and the coronary artery, distal to the blockage. The stent is self-expanding or uses a balloon to expand the stent in the heart wall. Various attachment means are provided to anchor the stent and prevent its migration. In one embodiment, a conduit is provided having a distal top which is more preferably a ball top, wire top, flare top or flip-down top. These top configurations anchor the shunt at one end in the coronary artery.