A fluid dispenser member for mounting on an opening (90) of a fluid reservoir (9), said member comprising: a body (1) inwardly defining a chamber (15) of variable volume, said body defining a bottom end (10); an actuator rod (7) that can be reciprocated axially in said body, and comprising a top portion defining a top end (79); a peripheral bearing collar (52) extending radially outwards, and including a bottom face (51) designed to come into abutment, at least indirectly, against an edge (911) of the opening (90) of the fluid reservoir; and a neck gasket (2) disposed in contact with the bottom face (51) of the collar so as to be compressed between the collar (52) and the edge (911) of the opening, the axial height HS1 axially separating the bottom face of the collar from the top end of the actuator rod being substantially equal to the axial height HI1 axially separating the bottom face of the collar from the bottom end of the body.