A gas inlet member of a CVD reactor includes a gas inlet housing having a gas distribution volume supplied with a process gas by a feed line and a multiplicity of gas lines, each formed as a tube and engaging openings of a gas outlet plate arranged in front of an inlet housing wall, and through which the process gas enters a process chamber. A coolant chamber adjoins the gas inlet housing wall and a coolant cools the gas inlet housing wall and outlet ends of the gas lines that are in heat-conductive contact with the gas inlet housing wall. The gas outlet plate is thereby thermally decoupled from the gas inlet housing wall such that the gas outlet plate, which is acted on by radiation heat coming from the process chamber, heats up more intensely than the outlet ends which extend into the openings of the gas outlet plate.