I found a video of Emmylou singing that on YouTube and verified that the tune in the DT is the same tune. (With a few variations, but that's only natural.)

I've been sitting here saying to myself, "I've played that! What IS it?"

It is the first part of a tune in O'Neill's Music of Ireland called 'Caitilin Tirriall' or 'Catherine Tyrrell'. In my edition it is #337. Number 336 is another setting of it, which seems to be similar, higher, and fancier.

I investigated the version in the DT, and words are nice but the MIDI has its quirks. I suggest finding MIDI's of 'Catherine Tyrrell' and seeing if they aren't more singable. Also, CT offers a little more variation.   

Dilydaly, I might find time to post chords. Are you still with us? What key do you like? Are you a soprano, alto, tenor or bass?

The song moves stepwise a lot. Playing chords to it will require subtlety.