Please substitute the following version of the Abstract, with changes shown by strikethrough (for deletions) or underlining (for added matter). The heat exchanger (I) is provided for a heating system with integrated fuel cells (Z) for the production of electricity and with an additional burner (B). Electrical and thermal energy (E, Q) can be produced in this heating system from a gaseous fuel or from a fuel brought into the gas form by means of the fuel cells, and/or thermal energy (Q) can be produced by means of the additional burner. One part of the thermal energy present in the form of hot exhaust gases can be transferred in the heat exchanger to a liquid heat transfer medium, in particular water or an oil. The heat transfer medium is provided for heat transport for the purpose of room heating and/or process water heating. The heat exchanger forms a compact unit which is also made of a material of good thermal conductivity. Two separate passages (12, 13) are arranged in the heat exchanger inside a double-walled jacket (10). The jacket has a structured inner space (11) which forms a communicating vessel. Due to the structuring the heat transfer medium flows through at least a predominant part of the inner space.