A machine for filling containers in which the filling rate for the products can be varied, thus enabling the filling machine to be used for different filling products, and in particular products having different viscosities. The machine comprises at least one filling station comprising a filling spout provided with a supply opening and a discharge opening, a blocking system mounted in the filling spout, and a supply pipe connected to the supply opening upstream of said blocking system. The supply pipe comprises at least one section made of a flexible tube, the supply means apparatus further comprising a clamp adapted to clamp the flexible tube in order to modify the cross-section of the internal passage of the flexible tube, the clamp adapted to move between at least two positions corresponding to different cross-sections of the internal passage of the flexible tube.