There are several different (and unrelated) songs based on the same old joke, most of which have already been mentioned; so far as I can tell, they are all modern compositions, though the one recorded by Harry Belafonte appears to have been modelled on an older song on a different subject. The one that people tend to imagine is Irish is the one written by Buffy Ste. Marie; she set it to a traditional tune known in Ireland (though also in America and other countries) which is probably where the misunderstanding crept in.