A machine for dispensing liquids, especially for filling cartons at a higher rate of speed, is disclosed. The machine includes three bellows arranged in vertical alignment, with upper and lower sleeves interposed between the bellows. The discharge nozzle is located below the lowest bellows. The upper sleeve moves axially relative to the nozzle to control the opening and closing of the discharge valve. The lower sleeve also moves axially to control the filling and recharging of the bellows directly above the discharge nozzle. The method of operating the components in timed relation to achieve effective and efficient carton filling, as well as draining and cleaning, is disclosed.