A process and a device are provided for separating carbon dioxide from a breathing gas mixture by means of a “Fixed Site Carrier” membrane. The breathing gas mixture is guided in the device on a side of a selective, semipermeable membrane, which is provided with amine groups, which are bound covalently to a polymer. Through the membrane, the transport of the components of the gas mixture can take place. The membrane is selected to be such that the permeability for CO2 is substantially higher than the permeability for the other gas components of the breathing gas mixture. The membrane has or is associated with means for guiding the gas, which acts to guide the gas mixture on one side along the membrane. The membrane separates volume areas in which different CO2 partial pressures prevail from one another.