A fluid dispenser pump including a first piston, a second piston, and a dispenser head with a dispenser orifice, for actuating the pump, a shutter arranged upstream from the dispenser orifice, the shutter movable between closed and open positions. The second piston is formed outside a hollow part and is slidable inside the dispenser head. The first piston is slidable inside the hollow part, the hollow part including an axial opening defined by a radial edge through which passes a stem portion of the shutter. The stem portion is between a proximal radial shoulder and a distal radial shoulder of the shutter, the shutter being moved from its closed position to its open position by the radial edge of the hollow part co-operating with the distal radial shoulder, and being moved from its open position to its closed position by the radial edge co-operating with the proximal radial shoulder.
