I found a version that's a bit longer and slightly different.

LET'S TAKE AN OLD FASHIONED WALK
(Irving Berlin, 1948)

VERSE 1
HE: Some couples go for a buggy ride
When they start caring a lot.
Others will bicycle side by side
Out to some romantic spot.
But when you haven't a sou,
There's only one thing to do.

CHORUS
Let's take an old fashioned walk;
I'm just bursting with talk—
What a tale could be told
If we went for an old fashioned walk!
Let's take a stroll through the park,
Down a lane where it's dark,
And a heart that's controlled
May relax on an old fashioned walk
I know for a couple who seemed to be miles apart,
There's nothing like walking and having a heart-to-heart
I know a girl who declined,
Couldn't make up her mind—
She was wrapped up and sold
Coming home from an old fashioned walk.

VERSE 2
SHE: I used to dream of a millionaire
Handsome and rich from the States,
Taking me out for a breath of air,
Saying, "The carriage awaits."
But since you haven't a sou
And I have nothing to do—

REPEAT CHORUS

This song appeared in the 1949 Musical Play, Miss Liberty. It was originally intended for the 1948 film Easter Parade, but it was replaced by "We're a Couple of Swells."

Source: The Complete Lyrics of Irving Berlin, edited by Robert Kimball and Linda Emmet (Alfred A.Knopf, New York, 2001) page 413

Here's a video: http://www.youtube.com/watch?v=Hj7wlLwd4o0