Connector between a reaction pipe and a cooling pipe and method for connecting a reaction pipe to a cooling pipe. The cooling pipe is a double pipe having an inner pipe, disposed in the extension of the reaction pipe, and an outer pipe that surrounds the inner pipe to form an intermediate cooling space between them. A supply chamber is connected to the cooling space for supplying coolant thereto. A transition piece, connected to the reaction pipe, has a fork-shaped end portion composed of an inner portion and an outer portion with heat-insulating material between them. The outer portion is secured to the base of the supply chamber coaxially relative to the longitudinal axis of the cooling pipe. The supply chamber, transition piece and half shells form a prefabricated unit. An inner tube section is welded to, and extends out of, the supply chamber and is aligned with the inner portion of the transition piece. The inner tube section has a thickness, inner diameter and outer diameter corresponding to those of the inner pipe. The thickness, inner diameter and outer diameter of the half shells correspond to those of the outer pipe. The axial length of the half shells is greater than the length of the part of the inner tube section extending out of the supply chamber.