FWIW, the movie was "Lili," not "Lily" or "Lilly," and the song title has the same spelling of the name.

HI LILI, HI LOW
(B Kaper, H Deutsch)

Chorus:
The song of love is a sad song,
Hi Lili, Hi Lili, Hi low.
The song of love is a song of woe -
Don't ask me how I know.
The song of love is a sad song,
For I have loved and it's so.
I sit by the window and watch the rain
Hi Lili, Hi Lili, Hi low.
Tomorrow I'll probably love again
Hi Lili, Hi Lili, Hi low.

In ev'ry tree there sits a bird
Singing a song of love,
In ev'ry tree ther sits a bird
And ev'ry one I ever heard
Would break my heart without a word
Singing a song of love.

(Chorus)