Process for cryogenic separation of a feed stream containing methane and air gases in which: - the feed stream is cooled in order to produce a cooled stream, - at least one portion of the cooled stream is sent to one level of a distillation column, - a bottom stream is drawn off from the distillation column, the bottom stream being enriched in methane relative to the feed stream, - a stream enriched in oxygen and in nitrogen relative to the feed stream is drawn off from the distillation column, - at least one noncombustible dilution stream that is more volatile than oxygen is introduced into the distillation column at at least one level lower than the one at which the cooled stream is introduced, characterized in that the dilution stream is extracted from the feed stream. Facility for producing biomethane by purification of biogases derived from non-hazardous waste storage facilities (NHWSF) implementing the process.