A method of producing a purified argon product stream in a cryogenic air separation unit, the method comprising the steps of: separating argon from an oxygen-argon containing stream within a lower pressure column of the cryogenic air separation unit using an argon rectification column arrangement disposed within the lower pressure column, the separation of the argon from the oxygen-argon containing stream producing an impure argon stream (97, 98) having between about and 4% and 25% of oxygen impurities; warming the impure argon stream against the purified argon product stream, a waste gas stream (172A, 172B) from a pressure swing adsorption system (154, 250), and a balancing air stream of purified, compressed air; pressurizing the warmed, impure argon stream; purifying the warm, pressurized, impure argon stream by introducing the warm, pressurized impure argon stream into the pressure swing adsorption system to produce the purified argon product stream and the waste gas stream containing nitrogen impurities; and recycling the waste gas stream from the pressure swing adsorption system to the argon rectification column arrangement; wherein the argon rectification column arrangement comprises an argon rejection column disposed within the lower pressure column and an argon condensing assembly disposed within the lower pressure column at a location above the argon rejection column; wherein the argon rejection column is configured to receive a portion of the ascending vapor from within the lower pressure column as the oxygen-argon containing stream, the waste gas stream, and all or a portion of an argon-rich reflux stream from the argon condensing assembly and produce two or more argon-rich vapor streams and a descending oxygen rich liquid stream which is released into the lower pressure column; and wherein the argon condensing assembly is configured to receive all or portion of one of the argon-rich vapor streams from the argon rejection column and condense the all or a portion of the one of the argon-rich vapor streams against an oxygen enriched liquid stream to produce the argon-rich reflux stream; and wherein the impure argon stream is either all or a portion of one of the argon-rich vapor streams or is a portion of the argon-rich reflux stream. separating argon from an oxygen-argon containing stream within a lower pressure column of the cryogenic air separation unit using an argon rectification column arrangement disposed within the lower pressure column, the separation of the argon from the oxygen-argon containing stream producing an impure argon stream (97, 98) having between about and 4% and 25% of oxygen impurities; warming the impure argon stream against the purified argon product stream, a waste gas stream (172A, 172B) from a pressure swing adsorption system (154, 250), and a balancing air stream of purified, compressed air; pressurizing the warmed, impure argon stream; purifying the warm, pressurized, impure argon stream by introducing the warm, pressurized impure argon stream into the pressure swing adsorption system to produce the purified argon product stream and the waste gas stream containing nitrogen impurities; and recycling the waste gas stream from the pressure swing adsorption system to the argon rectification column arrangement; wherein the argon rectification column arrangement comprises an argon rejection column disposed within the lower pressure column and an argon condensing assembly disposed within the lower pressure column at a location above the argon rejection column; wherein the argon rejection column is configured to receive a portion of the ascending vapor from within the lower pressure column as the oxygen-argon containing stream, the waste gas stream, and all or a portion of an argon-rich reflux stream from the argon condensing assembly and produce two or more argon-rich vapor streams and a descending oxygen rich liquid stream which is released into the lower pressure column; and wherein the argon condensing assembly is configured to receive all or portion of one of the argon-rich vapor streams from the argon rejection column and condense the all or a portion of the one of the argon-rich vapor streams against an oxygen enriched liquid stream to produce the argon-rich reflux stream; and wherein the impure argon stream is either all or a portion of one of the argon-rich vapor streams or is a portion of the argon-rich reflux stream.