It's by Allan Sherman and was on his album "My Son the Celebrity" (1963)
It's one of several songs in a medley called "Shticks of One and a Half a Dozen of the Other"

She wheels her wheelbarrow
Through streets that are narrow,
Her barrow is narrow, her hips are too wide.
So wherever she wheels it,
The neighborhood feels it,
Her girdle keeps scraping the homes on each side.

In Dublin's fair city,
Where girls are so pretty,
My Molly stands out 'cause she weighs 18 stone.
(That's 256 pounds.)
I don't mind her fat--but,
It's not only that--but,
She's cockeyed and muscle-bound, Molly Malone.

Bev and Jerry