The plant is used for producing oxygen by cryogenic air separation. The plant has a high-pressure column, a low-pressure column and a main condenser. An argon-elimination column is in fluid connection with an intermediate point of the low-pressure column and is connected to an argon-elimination column head condenser. An auxiliary column has a sump region, into which gas is introduced from the argon-elimination column head condenser. The head of the auxiliary column is connected to a return flow liquid line, in order to introduce a liquid stream from the high-pressure column or the head condenser. The liquid stream has an oxygen content which is at least equal to that of air. At least one part of the crude liquid oxygen from the sump of the high-pressure column is fed to the auxiliary column at a first intermediate point.