In a process for delivering pressurized gas from an apparatus for separating air by cryogenic distillation, a stream of oxygen-rich liquid or gas having a nominal flow rate is withdrawn from a low-pressure column, an oxygen-rich liquid purge stream is withdrawn as bottoms from the low-pressure column and the oxygen-rich liquid purge stream is sent to a storage tank, in the event of a reduction in the production by the column system or an increase in the demand by a customer, a back-up stream is withdrawn from the storage tank and vaporized in a back-up reboiler and only if the liquid level in the storage tank exceeds a given threshold, a liquid stream is withdrawn from the storage tank constituting at most 2% of the nominal production output of the oxygen-rich stream and is sent to the back-up reboiler.