Apparatus is described for use with a native heart valve of a subject, the apparatus including (1) a prosthetic valve support, comprising an upstream support portion, the upstream support portion having (a) a compressed configuration and an uncompressed configuration in which the upstream support portion has an inner perimeter that defines an opening; and (2) a prosthetic valve, advanceable into the opening defined by the upstream support portion, and intracorporeally couplable to the upstream support portion by being expanded within the opening defined by the upstream support portion, the apparatus being configured such that, when the prosthetic valve is expanded within the opening defined by the upstream support portion, the expansion of the prosthetic valve is restricted by the inner perimeter of the upstream support portion, without causing the prosthetic valve support to apply a radially-expansive force to the native annulus. Other embodiments are also described.