A constant flow rate valve has a housing in which a rotatable sleeve is positioned between the valve's inlet and outlet. Apertures in the sleeve move into alignment with the valve's inlet and outlet upon sleeve rotation to establish the fluid flow rate. A piston is movable longitudinally within the sleeve bore. The piston has openings which align with the apertures in the sleeve upon longitudinal motion of the piston. Pressure ducts connect each end of the sleeve to the inlet and outlet. The piston moves in response to relative pressure changes in the inlet and outlet communicated by the pressure ducts. The area of overlap between the piston's openings and the sleeve's apertures varies with piston motion to ensure a constant flow rate in spite of relative inlet and outlet pressure changes. The ends of the sleeve are in fluid communication to control the response sensitivity of the piston.