A prosthesis secures a replacement valve in a heart. The prosthesis includes a radially expandable inflow section and outflow section, and migration blocker rods. The inflow section has a tapered shape and is implanted within an atrium of a heart adjacent a native valve annulus. The outflow section couples to the inflow section, and is configured to be implanted through the native valve annulus and at least partially within a ventricle of the heart. The migration blocker rods extend circumferentially around at least a portion of the outflow section and hold native leaflets of the heart valve. In a contracted configuration, the prosthesis may be implanted through a catheter into the heart. In an expanded configuration, the tapered shape of the inflow section in the atrium cooperates with the migration blockers in the ventricle to hold the prosthesis against the native valve annulus.