@rrchapman, the tune is actually exactly the same. It is a Child ballad (# 56), which means it is a traditional tune Francis James Child wrote down in his studies of folklore. The tune is used in many traditional songs, including Star of the County Down. Which one came first is very hard to tell. "If you could hie to Kolob" is a Mormon hymn on the tune with distinctly mormon lyrics.
I was thinking I knew that tune! You're right: it's "Star of the County Down." Or maybe 'Star of the County Down' is 'Divers and Lazarus'. Sometimes it's hard to tell. Good call!
If that melody sounds hauntingly familiar, it's pretty much note-for-note the same as "Star of the County Down". (This is not a complaint -- it's common for folk melodies to be re-used.)