A system for implanting a heart valve, comprising: a radially self-expandable tubular body having an inflow end and an outflow end and a preformed groove disposed at an outer surface of the tubular body between the inflow end and the outflow end, the preformed groove extending at least partially around the tubular body and having a circumferential opening facing radially outward of the tubular body, and a plurality of projections extending in an axial direction of the tubular body and overlapping with the preformed groove, each projection having a first end and a second end, the first end being connected to the tubular body and the second end forming a free end unattached to the tubular body and arranged so as to overlap the preformed groove; a valve disposed within and attached to the tubular body; a fabric disposed around an outer circumference of tubular body and over second ends of the projections such that the fabric forms a pouch between the tubular body and the projections; and an elongate outer member configured to form a loop encircling the preformed groove so as to guide portions of native valve leaflets and/or chords into the preformed groove, wherein the outflow end of the tubular body includes a frustoconical shape that slopes radially outward from the preformed groove toward the outflow end when the outflow end but not the inflow end has been released from a delivery catheter, wherein when the outflow end but not the inflow end has been released from the delivery catheter the frustoconical shape slopes radially outward 10° - 20° with regard to a longitudinal center axis of the tubular body, and wherein when the outflow end but not the inflow end of the tubular body has been released from the delivery catheter, the elongate outer member is configured to be moved into the preformed groove by sliding the elongate outer member along a slope of the tubular body formed by the frustoconical shape in a direction moving radially inward along the slope of the tubular body from an outflow end of the tubular body toward an inflow end of the tubular body such that the elongate outer member is moved into the pouch, the native valve leaflets and/or chords are disposed between the tubular body and the elongate outer member and are trapped in the pouch. a radially self-expandable tubular body having an inflow end and an outflow end and a preformed groove disposed at an outer surface of the tubular body between the inflow end and the outflow end, the preformed groove extending at least partially around the tubular body and having a circumferential opening facing radially outward of the tubular body, and a plurality of projections extending in an axial direction of the tubular body and overlapping with the preformed groove, each projection having a first end and a second end, the first end being connected to the tubular body and the second end forming a free end unattached to the tubular body and arranged so as to overlap the preformed groove; an inflow end and an outflow end and a preformed groove disposed at an outer surface of the tubular body between the inflow end and the outflow end, the preformed groove extending at least partially around the tubular body and having a circumferential opening facing radially outward of the tubular body, and a plurality of projections extending in an axial direction of the tubular body and overlapping with the preformed groove, each projection having a first end and a second end, the first end being connected to the tubular body and the second end forming a free end unattached to the tubular body and arranged so as to overlap the preformed groove; a valve disposed within and attached to the tubular body; a fabric disposed around an outer circumference of tubular body and over second ends of the projections such that the fabric forms a pouch between the tubular body and the projections; and an elongate outer member configured to form a loop encircling the preformed groove so as to guide portions of native valve leaflets and/or chords into the preformed groove, wherein the outflow end of the tubular body includes a frustoconical shape that slopes radially outward from the preformed groove toward the outflow end when the outflow end but not the inflow end has been released from a delivery catheter, wherein when the outflow end but not the inflow end has been released from the delivery catheter the frustoconical shape slopes radially outward 10° - 20° with regard to a longitudinal center axis of the tubular body, and wherein when the outflow end but not the inflow end of the tubular body has been released from the delivery catheter, the elongate outer member is configured to be moved into the preformed groove by sliding the elongate outer member along a slope of the tubular body formed by the frustoconical shape in a direction moving radially inward along the slope of the tubular body from an outflow end of the tubular body toward an inflow end of the tubular body such that the elongate outer member is moved into the pouch, the native valve leaflets and/or chords are disposed between the tubular body and the elongate outer member and are trapped in the pouch.