A method and apparatus for producing a purified liquid argon product in which liquid argon having oxygen impurities is produced in a cryogenic air separation plant by separating argon from oxygen within an argon column. An impure liquid argon stream, composed of part of the liquid argon, is purified in an adsorbent bed by adsorbing the oxygen impurities in an adsorbent to produce a purified liquid argon stream that constitutes the purified liquid argon product. During adsorption, the adsorbent bed is maintained at a reduced temperature with a coolant to prevent vaporization of the liquid argon. The bed is then regenerated by draining residual liquid argon from the adsorbent bed, introducing the residual liquid argon back into the air separation plant and then desorbing the oxygen impurities with a regeneration gas. After regeneration, the adsorbent bed is refilled with purified liquid argon prior to being brought back on-line.