This device comprises an elongated member (10), which includes a distal part (11) provided with a through hole (12) joining first and second openings (15, 16) respectively defined in radially opposed first and second sides (13, 14) of the member, and an elongated tubular body (20) which is axially moveably disposed around the member (10) between a load position and a cut position. According to the invention, the tubular body is mounted around the member with a non-zero radial tolerance (T). Besides, between the load position and the cut position, the tubular body is adapted to be in an intermediate position in which, on the first side (13) of the member, the first opening (15) is kept at least partly free by the tubular body, whereas, on the second side (14) of the member, the second opening (16) is axially covers in totality by the tubular body and a radial gap (24), through which a suture (4) passes, is defined between a distal end (21) of the tubular body and a distal end (17) of the member, so that, when the tubular body passes from the intermediate position to the cut position, the gap is radially enlarged by a biasing action of the suture between the tubular body and the second side of the member, up to correspond to the tolerance, while the distal end of the tubular body radially moves towards the first side of the member, up to cooperate without clearance with the hole for cutting the suture.
