A fluid dispenser device comprising: a reservoir (1) including a neck (12); a dispenser member (2) that is mounted in the neck (10), the dispenser member (2) being provided with a dip tube (23); and a cover tube (5) that surrounds said dip tube (23); the fluid dispenser device being characterized in that, in the proximity of its connection top end (52), said cover tube (5) includes at least one vent (55) having a total flow section (St) that presents a size that is sufficient to enable the fluid to rise in the cover tube (5) while the cover tube is being inserted quickly into the fluid reservoir (1) in the filled state, without being significantly disturbed by the air that is evacuated from the cover tube (5) through said at least one vent (55).