A method and apparatus of producing high purity oxygen in connection with low purity liquid oxygen produced by a plurality of cryogenic air separation plants. The low purity liquid oxygen from the air separation plants is introduced into a distillation column of an auxiliary cryogenic rectification plant that is reboiled by nitrogen also produced by the air separation plants. Nitrogen is separated from the low purity liquid oxygen to produce high purity liquid oxygen from residual liquid located in a bottom region of the distillation column that can be taken as a product. Optionally, an argon column can be connected to the distillation column to produce a liquid argon product stream.