A Side: mx 80967 (label & runout); 2-G-62 (stamped under label).
B Side: mx 80968 (label & runout); 2-G-52 (stamped under label).
Both recorded New York, NY, Apr. 14, 1923.
The Talking Machine World, June 15, 1923, page 190: Advance Record Bulletins for July, 1923 [i.e., released in late June 1923].
Images
Number:858872 THUMBNAIL Uploaded By:xiphophilos Description: Columbia A3876 A Side Label
Number:858873 Uploaded By:xiphophilos Description: Columbia A3876 B Side Label
Number:1066289 Uploaded By:xiphophilos Description: Columbia A3876 A Side Label Font Variant
Number:1066290 Uploaded By:xiphophilos Description: Columbia A3876 B Side Label Font Variant
W.B.lbl has now told me that both records were pressed at Columbia's Bridgeport, CT plant. The narrower capital letters 'C', 'D', and 'O' in the "Duet with Orchestra accompaniment" line on my older pics are from a pre-1920 Linotype font (7 point Gothic No. 4), whereas the wider 'C', 'D' and 'O' on the recently uploaded label are from post-1920. Columbia's typesetters continued to use the older types for a long time, some into the 1950s, and mixed different types of the same font as well.
"Barney Google" (after the then popular comic strip) must have been a particular hit of the Hare-Jones duo. They recorded the song also for Edison, Emerson, and Brunswick.