Fibula has correctly identified the song. (There are several other songs with the same title.) This one is also on Padraigin Ni Uallachain's album "An Irish Lullaby," Shanachie CD #78036, 2000.

On "A Prairie Home Companion," Cathal McConnell and Dave Richardson of The Boys of the Lough sang THE WILLOW TREE. You can hear the archived radio program at this page. The song begins 14 minutes and 15 seconds from the beginning of Segment 3.

Here's my transcription. I had to guess at the spelling of some of the place names.

THE WILLOW TREE

1. Down by the river there's a tall willow tree
Who weeps all night for you and me.

CHORUS: And it's lay down low, my love.
And it's lay you down, my own true love.
[It's] shadows are falling and night has come,
And it's lay down low, my love.

2. Under the branches of the brown thorn tree,
The wild bird is watching over Lissonshee (?).

3. On Lake Derravaragh there's a white-feathered swan
Who sings with sorrow the whole night long.

4. The moon is moving over Powmabrone (?)
Where two lovers are lying on their bed of stone.

5. Down by the river there's a tall willow tree
Who weeps all night for you and me.

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From Shlomo Caine: Joe, for what it's worth.... in Patty's song you had the place name "Pawmabrone" with a question mark... The correct place name is Poll na Brón which is in the Burren in county Clare.