A multi-zone jacketed pipe reactor (2; 60; 90; 130) for carrying out exothermic gaseous phase reactions and with at least one reaction zone (I) working with vaporisation cooling, at least one reaction zone (II) working with circulation cooling and, possibly, with additional zones (III, IV) is characterised in that one reaction zone (I) working with vaporisation cooling forms the first reaction zone to which is connected an additional reaction zone (II) working with circulation cooling. In this way there occurs at the beginning of the reaction, when the latter is most violent, very intensive cooling at a precisely controllable temperature and especially as well a temperature that is constant across the entire cross-section of the reactor while subsequently in a subsequent reaction zone working with circulating cooling by means of global counter-flow guidance of the heat transfer agent a constant cooling of the reaction gas is achieved.