A filling machine includes a filling tube passing through a spout endpiece to delimit with the spout endpiece, part of an air return circuit. The filling tube is vertically movable and delimits an exit passage for the liquid. A control circuit is provided for moving the filling tube to a low position and driving an obturation device to open the exit passage, putting the air return circuit, when the exit passage of the filling tube is immersed in the liquid, to atmospheric pressure, moving the filling tube towards its leveling position, putting the air return circuit in communication with a pressurized chamber, and bringing the filling tube to a leveling position.