An assembly for implanting a balloon-expandable prosthetic aortic heart valve in a patient, comprising: a delivery apparatus comprising an elongated shaft having an inflatable balloon, and a prosthetic aortic heart valve comprising a balloon-expandable frame having a radially collapsed and a radially expanded state, the frame comprising: an inflow row of openings at an inflow end portion of the frame, an outflow row of openings at an outflow end portion of the frame; and at least one intermediate row of openings between the inflow row of openings and outflow row of openings; wherein the outflow row of openings is formed by a circumferentially extending upper row of angled struts arranged end-to-end and a circumferentially extending intermediate row of angled struts arranged end-to-end, the upper and intermediate rows of angled struts (28, 32) interconnected by a plurality of axially extending struts and a plurality of angularly spaced, axially extending commissure window frame portions, wherein an upper end of each axially extending strut and an upper end of each commissure window frame portion is connected to a junction defined by the convergence of the lower ends of two angled struts of the upper row of angled struts, and a lower end of each axially extending strut and a lower end of each commissure window frame portion is connected to a junction defined by the convergence of the upper ends of two angled struts of the intermediate row of angled struts; and wherein the plurality of commissure window frame portions each comprise an enclosed opening between first and second axially oriented side struts, the prosthetic aortic heart valve further comprising: an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, and a leaflet structure positioned within the frame, the leaflet structure comprising a plurality of leaflets and a plurality of commissure portions that extend outwardly through a respective commissure window frame portion, wherein each of the plurality of leaflets comprises a lower edge having a scalloped shape which is sutured to the inner skirt between respective commissures of the leaflet structure. a delivery apparatus comprising an elongated shaft having an inflatable balloon, and a prosthetic aortic heart valve comprising a balloon-expandable frame having a radially collapsed and a radially expanded state, the frame comprising: an inflow row of openings at an inflow end portion of the frame, an outflow row of openings at an outflow end portion of the frame; and at least one intermediate row of openings between the inflow row of openings and outflow row of openings; an inflow row of openings at an inflow end portion of the frame, an outflow row of openings at an outflow end portion of the frame; and at least one intermediate row of openings between the inflow row of openings and outflow row of openings; wherein the outflow row of openings is formed by a circumferentially extending upper row of angled struts arranged end-to-end and a circumferentially extending intermediate row of angled struts arranged end-to-end, the upper and intermediate rows of angled struts (28, 32) interconnected by a plurality of axially extending struts and a plurality of angularly spaced, axially extending commissure window frame portions, wherein an upper end of each axially extending strut and an upper end of each commissure window frame portion is connected to a junction defined by the convergence of the lower ends of two angled struts of the upper row of angled struts, and a lower end of each axially extending strut and a lower end of each commissure window frame portion is connected to a junction defined by the convergence of the upper ends of two angled struts of the intermediate row of angled struts; and wherein the plurality of commissure window frame portions each comprise an enclosed opening between first and second axially oriented side struts, the prosthetic aortic heart valve further comprising: an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, and a leaflet structure positioned within the frame, the leaflet structure comprising a plurality of leaflets and a plurality of commissure portions that extend outwardly through a respective commissure window frame portion, wherein each of the plurality of leaflets comprises a lower edge having a scalloped shape which is sutured to the inner skirt between respective commissures of the leaflet structure. an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, and a leaflet structure positioned within the frame, the leaflet structure comprising a plurality of leaflets and a plurality of commissure portions that extend outwardly through a respective commissure window frame portion, wherein each of the plurality of leaflets comprises a lower edge having a scalloped shape which is sutured to the inner skirt between respective commissures of the leaflet structure.