This method makes it possible to fabricate a stopper (1) starting from a tubular skirt (12) suitable for surrounding a container neck, and provided both with retainer means for retaining it permanently around the neck, and also with fastener means for fastening it removably to the neck. In this method, the skirt and a blade (110) are moved in rotation relative to each other about the axis (X-X) of the skirt so that the blade cuts a peripheral line of weakness in the skirt, which line of weakness is made up firstly of through notches and secondly, between said notches around the periphery of the skirt, of breakable bridges that interconnect, in the same direction as the axis, a non-removable portion of the skirt, which portion is provided with the retainer means, and a removable portion of the skirt, which portion is provided with the fastener means. With a view to improving the method simply and inexpensively, provision is made for the skirt and the blade to be driven relative to each other through an angular stroke that is strictly greater than 360°, while progressively modifying the relative axial position between the skirt and the blade by a movement in axial translation (T). The line of weakness is thus in the shape of a helix having its two peripheral ends defining between them a non-breakable strip that permanently interconnects the removable and the non-removable portions of the skirt.