A connection between an uncooled pipe and a cooled double-wall pipe with between the walls a hollow space run through by cooling fluid let into the hollow space and taken from the hollow space through side passages connected to cooling fluid inlet and outlet ducts and including a connection zone with longitudinal cross section generally in fork form to have single tubular end connected to the uncooled pipe and an opposite double-wall tubular end to which are connected the corresponding walls of the cooled double-wall pipe so that the fork bottom constitutes a terminal closing of the hollow space and a side passage is made opposite the connection and a portion thereof nearest the uncooled pipe is almost at the height of the fork bottom. A double-pipe heat exchanger includes such a connection.