The invention relates to a reactor for catalytic conversion of a gas mixture, in particular for catalytic ammonia synthesis from a gas mixture substantially comprising nitrogen and hydrogen, having a pressure vessel (2) which has an inlet (3) for the gas mixture and an outlet (4) for the reaction product, having a first catalyst bed (10), a second catalyst bed (20), and a third catalyst bed (30) which are arranged such that the gas mixture introduced through the inlet (4) is first directed through the first catalyst bed (10), then through the second catalyst bed (20) and finally through the third catalyst bed (30) to the outlet (4), having a first heat exchanger (11) for cooling the gas mixture flowing out of the first catalyst bed (10) and having a second heat exchanger (21) for cooling the gas mixture flowing out of the second catalyst bed (20), wherein the reactor (1) has a third heat exchanger (31) for cooling the gas mixture flowing out of the third catalyst bed (30) in the direction of the outlet (4), the first catalyst bed (10), the second catalyst bed (20) and the third catalyst bed (30) are arranged in a cylindrical reaction region (2.1) of the pressure vessel (2), and the pressure vessel (2) has a central pipe (50) extending along a cylinder axis (Z) of the reaction region (2.1), via which central pipe the inlet (3) of the pressure vessel (2) is connected to the third heat exchanger (31).