The pumping device comprises: a dry rough vacuum pump provided with a gas inlet opening connected to a vacuum chamber and moreover has a gas outlet opening leading into a pipe; a delivery check valve placed in the pipe at the outlet of the dry rough vacuum pump; and an ejector mounted in parallel relative to the delivery check valve. The pumping method includes the following steps: pumping the gases, contained in the vacuum chamber, by means of the dry rough vacuum pump through the gas inlet opening; connecting the gas outlet opening of the dry rough vacuum pump to an ejector; measuring the electrical power used by the dry rough vacuum pump and the pressure of the gases in the pipe at the outlet of the dry rough vacuum pump; setting in motion the ejector, after a time delay, when the pressure of the gases at the outlet of the dry rough vacuum pump has exceeded a rising edge set value and when the electrical power used by the dry rough vacuum pump exceeds a rising edge set value; stopping the ejector when the electrical power used by the dry rough vacuum pump exceeds a falling edge set value and when the pressure of the gases in the pipe at the outlet of the dry rough vacuum pump exceeds a falling edge set value.