Systems and methods for replacing a native heart valve. An anchor comprises multiple coils adapted to support a heart valve prosthesis. At least one of the coils is normally at a first diameter, and is expandable to a second, larger diameter upon application of radial outward force from within the anchor. An expansible heart valve prosthesis is provided and is configured to be delivered into the anchor and expanded inside the multiple coils. This moves at least one coil from the first diameter to the second diameter while securing the anchor and the heart valve prosthesis relative to each other. The system further includes a seal on the anchor that can prevent at least some blood leakage after implantation of the heart valve prosthesis in the helical anchor. Additional apparatus and methods are disclosed.