An assembly for implanting a balloon-expandable prosthetic aortic valve in a patient's body, the assembly comprising: a delivery apparatus comprising an elongated shaft having an inflatable balloon, and a radially expandable prosthetic aortic valve adapted to be mounted on the shaft in a radially collapsed configuration for delivery into the body, the prosthetic aortic valve comprising: a balloon-expandable annular frame made of a nickel-cobalt-chromium alloy and having an inflow end and an outflow end, wherein when the prosthetic aortic valve is collapsed from the expanded configuration to the collapsed configuration, the axial length of the frame increases, a leaflet structure positioned within the frame; and an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, characterized by the inner skirt comprising a weave of a first set of strands with a second set of strands, wherein (i) both the first and second sets of strands are non-parallel with the axial direction of the prosthetic aortic heart valve or (ii) the first and second sets of strands are elastic, and one of the first and second sets of strands is parallel and the other one of the first and second sets of strands is perpendicular to the axial direction of the prosthetic aortic heart valve; wherein the inner skirt has an inflow edge oriented toward the inflow end of the frame, an outflow edge oriented toward the outflow end of the frame, opposing oblique edges (86, 88) that are non-perpendicular to the inflow edge, and opposing edge portions (90, 92) that are sewn together with a suture line that is parallel to the oblique edges (86, 88). a delivery apparatus comprising an elongated shaft having an inflatable balloon, and a radially expandable prosthetic aortic valve adapted to be mounted on the shaft in a radially collapsed configuration for delivery into the body, the prosthetic aortic valve comprising: a balloon-expandable annular frame made of a nickel-cobalt-chromium alloy and having an inflow end and an outflow end, wherein when the prosthetic aortic valve is collapsed from the expanded configuration to the collapsed configuration, the axial length of the frame increases, a leaflet structure positioned within the frame; and an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, characterized by the inner skirt comprising a weave of a first set of strands with a second set of strands, wherein (i) both the first and second sets of strands are non-parallel with the axial direction of the prosthetic aortic heart valve or (ii) the first and second sets of strands are elastic, and one of the first and second sets of strands is parallel and the other one of the first and second sets of strands is perpendicular to the axial direction of the prosthetic aortic heart valve; wherein the inner skirt has an inflow edge oriented toward the inflow end of the frame, an outflow edge oriented toward the outflow end of the frame, opposing oblique edges (86, 88) that are non-perpendicular to the inflow edge, and opposing edge portions (90, 92) that are sewn together with a suture line that is parallel to the oblique edges (86, 88). a balloon-expandable annular frame made of a nickel-cobalt-chromium alloy and having an inflow end and an outflow end, wherein when the prosthetic aortic valve is collapsed from the expanded configuration to the collapsed configuration, the axial length of the frame increases, a leaflet structure positioned within the frame; and an annular inner skirt positioned within the frame, the inner skirt being secured to the inside of the frame, characterized by the inner skirt comprising a weave of a first set of strands with a second set of strands, wherein (i) both the first and second sets of strands are non-parallel with the axial direction of the prosthetic aortic heart valve or (ii) the first and second sets of strands are elastic, and one of the first and second sets of strands is parallel and the other one of the first and second sets of strands is perpendicular to the axial direction of the prosthetic aortic heart valve; wherein the inner skirt has an inflow edge oriented toward the inflow end of the frame, an outflow edge oriented toward the outflow end of the frame, opposing oblique edges (86, 88) that are non-perpendicular to the inflow edge, and opposing edge portions (90, 92) that are sewn together with a suture line that is parallel to the oblique edges (86, 88).