That's always been one of my very favourite early Stones singles. I didn't realise they didn't write it until now. Apparently the DJ Murray the K gave them the record and suggested they record it, and they did so at the Chess Studios in Chicago at the end of a less than spectacular first US tour.
Although Lowe offered it to Cash before he recorded it himself. I seem to remember reading that Johnny didn't want to do it, but that Rick Rubin persuaded him otherwise.