Pump for dispensing a fluid product, having a piston having a pump chamber and an inlet and outlet valve. The outlet valve has an outlet valve element that slides inside the pump chamber. The pump chamber has a passage so that, at the end of actuation, the outlet valve element collaborates non-sealingly with the passage to open the outlet valve to expel product. The inlet valve has an inlet valve element sliding in a sleeve containing a spring pressing against the inlet valve element and against an end wall of the sleeve. The outlet valve element is formed by a component inserted with the ability to move into the piston, with the interposition of a second spring, the force exerted by the second spring is greater than the force exerted by the spring in the rest position, and less than the force exerted by the spring in the actuated position.