Thank you very much, Braveharp. Now, I hope I can figure out who asked me for that song back in 2004. You didn't have the verse about the "fair maid, fair maid"??
Here are the lyrics, without the tune, for easier reading:

The Fairy Ring

As Katie one night by moon light bright
sat busy spinning her thread.
'Twas Tim came by with a laugh in his eye
and now listen to what he said:
I've had a fling in a fairy ring
where fairies dance and play and sing
Won't you come this night while stars are bright
you'll see, you'll see
for I'm the lad the fairies have had
to tea, to tea

As Katy one night by moon light bright
sat busy spinning her thread.
'Twas Tim came by with a laugh in his eye
and now listen to what he said:
I've had a fling in a fairy ring
where fairies dance and play and sing-
won't you come this night while stars are bright
you'll see, you'll see
for I'm the lad the fairies have had
to tea, to tea.