Thanks, giles.
Yes, that's the basic tune that Dan sings.

But as you can hear from the verse Claire posted from Dan's album "The First Christmas Morning," there's a little more nuance and variation to the tune.   The second two lines of the verse do not have exactly the same tune as the first two.

Thanks so much for that link, Bear.   Dan's voice is so hauntingly beautiful on that song it makes me even sadder that he died so prematurely this past year.    But I'm so glad I have his Christmas album.

Genie