This progressing cavity pump includes a helical rotor (2) mounted to turn inside a helical stator (3). The stator (3) and the rotor (2) are disposed such that the cavities (4) formed therebetween move from the inlet (5) towards the outlet (6). In this cavity pump, hydraulic regulation (HR) means are provided for obtaining internal recirculation of the pumped fluid between at least two of the cavities (4) under conditions capable of performing at least one function selected from: achieving the desired pressure distribution along the pump, stabilizing the temperatures, controlling the leakage flow rates, and compensating for the volumes of compressed gas.