A non-drip device for a bottle is in the form of an outwardly projecting peripheral lip. This peripheral lip has a triangular section of thickness that decreases out to an outer tip, the lip being defined by a top side and by a bottom side that, between them, form an angle lying in the range 5° to 45° . This outwardly projecting peripheral lip has a length advantageously lying in the range 0.1 mm to 0.8 mm. It may be provided at the top end of the skirt of a cover of the engagement type. In a variant embodiment, this lip may be formed integrally at the neck of the bottle. Such a lip structure makes it possible, at the time of pouring, and in effective manner, to avoid formation of drops that run down along the outside face of the bottle.