From The Lester S. Levy Collection of Sheet Music:

MY IRISH MOLLY O
Words, William Jerome. Music, Jean Schwartz. 1902
"As sung by Miss Blanche Ring of the Frank Daniels Comic Opera Co.
in Chas. B. Dillingham's Production of the Musical Farce 'Sergeant Brue' "

Molly dear, and did you hear the news that's going round?
Down in the corner of my heart a loving place you've found,
And ev'ry time I gaze into your Irish eyes of blue,
They seem to whisper, "Darling boy, my love is all for you."

CHORUS: Molly, my Irish Molly, my sweet acushla dear,
I'm fairly off my trolley, my Irish Molly, when you are near.
Spring time, you know, is ring time. Come, dear, don't be so slow.
Change your name. G'wan, be game, begorra, and I'll do the same,
My Irish Molly, O!

Molly dear, and did you hear? I furnished up a flat:
Three little cosy rooms and bath with "Welcome" on the mat;
Ten dollars down and two a week, I'll soon be out of debt.
It's all complete except they haven't brought the cradle yet.

[G'wan = go on.]
[This is very similar to the unattributed MY IRISH MOLLY-O in the Digital Tradition, but the DT has an extra verse that isn't in the sheet music.]