Apparatus, comprising: a frame assembly that comprises: a valve body that circumscribes a longitudinal axis and defines a lumen along the axis; a plurality of upstream arms that are coupled to the valve body at a first axial level with respect to the longitudinal axis, each of the arms extending radially outward from the valve body to a respective arm-tip; and a plurality of downstream legs that are coupled to the valve body at a second axial level with respect to the longitudinal axis, and that extend radially outward from the valve body and toward the plurality of arms; a tubular liner that lines the lumen, and that has an upstream end and a downstream end; a plurality of prosthetic leaflets, disposed within the lumen, attached to the liner, and arranged to facilitate one-way upstream-to-downstream fluid flow through the lumen, the first axial level being upstream of the second axial level; a first sheet of flexible material, the first sheet having (i) a greater perimeter, and (ii) a smaller perimeter that defines an opening, the first sheet being attached to the plurality of arms with the opening aligned with the lumen of the valve body; and a second sheet of flexible material: the second sheet having a first perimeter and a second perimeter, the first perimeter being attached to the greater perimeter of the first sheet around the greater perimeter of the first sheet, the second sheet extending from the first perimeter radially inwards and downstream toward the second perimeter, the second perimeter circumscribing, and attached to, the valve body at a third axial level that is downstream of the first axial level, wherein: the first sheet, the second sheet, and the liner define an inflatable pouch therebetween, the inflatable pouch defining an interior space therein, the first sheet defining an upstream wall of the pouch, the second sheet defining a radially-outer wall of the pouch, and the liner defining a radially-inner wall of the pouch, and characterized in that: each of the legs presses the second sheet into contact with the first sheet. a frame assembly that comprises: a valve body that circumscribes a longitudinal axis and defines a lumen along the axis; a plurality of upstream arms that are coupled to the valve body at a first axial level with respect to the longitudinal axis, each of the arms extending radially outward from the valve body to a respective arm-tip; and a plurality of downstream legs that are coupled to the valve body at a second axial level with respect to the longitudinal axis, and that extend radially outward from the valve body and toward the plurality of arms; a valve body that circumscribes a longitudinal axis and defines a lumen along the axis; a plurality of upstream arms that are coupled to the valve body at a first axial level with respect to the longitudinal axis, each of the arms extending radially outward from the valve body to a respective arm-tip; and a plurality of downstream legs that are coupled to the valve body at a second axial level with respect to the longitudinal axis, and that extend radially outward from the valve body and toward the plurality of arms; a tubular liner that lines the lumen, and that has an upstream end and a downstream end; a plurality of prosthetic leaflets, disposed within the lumen, attached to the liner, and arranged to facilitate one-way upstream-to-downstream fluid flow through the lumen, the first axial level being upstream of the second axial level; a first sheet of flexible material, the first sheet having (i) a greater perimeter, and (ii) a smaller perimeter that defines an opening, the first sheet being attached to the plurality of arms with the opening aligned with the lumen of the valve body; and a second sheet of flexible material: the second sheet having a first perimeter and a second perimeter, the first perimeter being attached to the greater perimeter of the first sheet around the greater perimeter of the first sheet, the second sheet extending from the first perimeter radially inwards and downstream toward the second perimeter, the second perimeter circumscribing, and attached to, the valve body at a third axial level that is downstream of the first axial level, the second sheet having a first perimeter and a second perimeter, the first perimeter being attached to the greater perimeter of the first sheet around the greater perimeter of the first sheet, the second sheet extending from the first perimeter radially inwards and downstream toward the second perimeter, the second perimeter circumscribing, and attached to, the valve body at a third axial level that is downstream of the first axial level, the first sheet, the second sheet, and the liner define an inflatable pouch therebetween, the inflatable pouch defining an interior space therein, the first sheet defining an upstream wall of the pouch, the second sheet defining a radially-outer wall of the pouch, and the liner defining a radially-inner wall of the pouch, and characterized in that: each of the legs presses the second sheet into contact with the first sheet.