A fluid dispenser valve including a valve body extending along a longitudinal axis between a first axial end and a second axial end, the valve body containing a metering chamber; and a valve member that slides axially in the valve body between a rest position and a dispensing position, for selectively dispensing the contents of the metering chamber. The metering chamber is connected, in the valve member rest position, to a fluid reservoir, to fill the metering chamber by gravity when the valve is upsidedown with the metering chamber below the reservoir while the valve member is in the rest position, and to empty the metering chamber by gravity when the valve is upright. The valve body includes a first hole and a second hole, to fill and to empty the metering chamber, the first hole being axially offset along the longitudinal axis relative to the at second hole.