The electrolysis of aqueous alkali metal chloride solutions under pressure has already been carried out. In accordance with the invention there is applied a pressure of at least 8 bars in the anode space for such an electrolysis, as well as a special process for the work-up of the products of the anode space. The mixture of gas and anolyte leaving the anode space is at first mechanically separated, and the anolyte is then depressurized into a stripping column. In the course of this process a dechlorination and cooling of the anolyte are taking place. The anolyte is to enter the pressure release vessel (stripping column) with a temperature being above the boiling point of the anolyte at atmospheric pressure. The reduced pressure is to be between 2 bars and atmospheric pressure. By means of these measures it may be ensured that the anolyte in the stripping column reaches boiling point. Finally the chlorine set free in the pressure release operation is separated from the anolyte.