According to this process, there is supplied to the passengers a first fraction of air enriched in oxygen from independent supply elements, in particular high pressure cylinders (16), during a descent phase of the aircraft between a normal cruising altitude and an intermediate re-routing altitude. There is moreover compressed air taken from a source of compressed air belonging to the aircraft, to produce (in 2) a second fraction of the air enriched in oxygen which is delivered to the passengers, at least during a phase of stabilized flight of the aircraft, adjacent the re-routing altitude, greater than 5,500 meters.