This is a highly-simplified version which I have lowered in key to suit my voice (and to simplify the chords), but the chords used in the movie "Gilda" are:
C#m F#m G#
Here is how I like to play it (so that it's in my vocal range):
Gm Cm
When Mrs O'Leary's cow kicked the lantern
D Gm
In Chicago town
Gm Cm
They say that started the fire
D D or D7
That burned Chicago down
Cm Gm
That's the story that went around
Gm D
But here's the real low-down
Gm D Gm D
Put the blame on Mame boys
Gm D Gm
Put the blame on Mame
Cm Gm
Mame kissed a buyer from out of town
Gm Gm D
That kiss burned Chicago down so you can
Gm D Gm D
Put the blame on Mame boys
Gm D Gm
Put the blame on Mame
(and so on with the rest of the verses...this version is bare-bones and it could really be punched up with some simple Tab melodies and jazz chords or something)
Remember the blizzard back in Manhattan
In eighteen-eighty-six
They say that traffic was tied up
And folks were in a fix
That's the story that went around
But here's the real low-down
Put the blame on Mame boys
Put the blame on Mame
Mame gave a chump such an ice-cold No
For seven days they shovelled snow
So you can put the blame on Mame boys
Put the blame on Mame
When they had the earthquake in San Francisco
Back in nineteen-six
They said that Mother Nature
Was up to her old tricks
That's the story that went around
But here's the real low-down
Put the blame on Mame boys
Put the blame on Mame
One night she started to shim and shake
That brought on the Frisco quake
So you can put the blame on Mame boys
Put the blame on Mame
They once had a shootin' up in the Klondike
When they got Dan McGrew
Folks were putting the blame on
The lady known as Lou
That's the story that went around
But here's the real low-down
Put the blame on Mame boys
Put the blame on Mame
Mame did a dance called the hoochy-coo
That's the thing that slew McGrew
So you can put the blame on Mame boys
Put the blame on Mame