A process for producing a mat with thin walled plastic tubes with a dia. of up to 5 mm (capillary tubes), which branch off radially from main tubes that supply a heating / cooling medium, comprises drilling holes with the dia of the inner plastic tubes, in the main tubes, and welding by the application of heat and pressure. The plastic pipes (3) are heated over their end regions (4) so the inner dia increases. Use - The process is used to produce a thin walled plastic tube heating or cooling mat. Advantage - The pipe flow cross-section is not reduced by the welding process.