The present invention is directed to a process for simultaneously obtaining pure carbon monoxide and pure hydrogen in a steam reformer plant for hydrogen or ammonia generation, having a primary reformer, a secondary reformer and downstream thereof, a CO conversion stage. A part gas stream of the synthesis gas stream, which is discharged from the secondary reformer having a CO content of between 2 and 20 mol. % and is at a temperature of from 200 to 500.degree. C. and a pressure between 15 and 50 bar, is removed between the secondary reformer and the CO conversion stage. The part gas stream is then is cooled to a temperature below 100.degree. C., thereby condensing out the major part of the steam contained in the gas stream. The remaining raw synthesis gas stream is then guided by way of a multistage gas separation plant in which the gas components H.sub.2, residual steam, CH.sub.4, CO.sub.2 and optionally N.sub.2 are separated, either individually or together, from the CO. The separated gas components are then compressed to a pressure which exceeds the pressure in the CO conversion stage and are recombined to form a mixed gas stream. The mixed gas stream is then guided into the CO conversion stage of the steam reformer after being heated to a temperature of from 200 to 500.degree. C. The remaining pure CO fraction is removed in separate manner and may optionally, be supplied for further processing.