The method provides for arranging a pick-up member for the tubular knitted article (M) coaxially with the needle cylinder, when knitting has been completed. The pick-up member (23) is provided with a plurality of pick-up hooks (25A, 25B) arranged according to a circular ring subdivided into two circular half-rings that can be overturned over each other. The tubular article is engaged by the pick-up hooks while the stitches of the last course of the knitted article are still held by the needles (5) of the knitting machine. The stitches of the last curse are then removed from the needles and the tubular article is removed from the needle cylinder (3) by means of the pick-up member (23). The two half-rings of pick-up hooks are overturned one onto the other, putting one of two opposite edges of the toe of the tubular article onto the other. The two edges are then sewn or linked.