The song is called "How do you Do?"
These are the lyrics from the film the attraction was based off of, Song of the South:
How Do You Do?
Words and Music by Robert MacGimsey
Performed by James Baskett & Johnny Lee
(and other uncredited characters)
© 1944 Walt Disney Music Company
How do you do? Fine! A friendly greetin'.
How do you do? Say it when you're meetin'.
How do you do? With everyone repeatin':
Pretty good, sure as you're born.
What goes up is sure to come down,
Penny lost is a penny found,
I'll "howdy" you, you'll "howdy" back,
This for that an' tit for tat.
Chorus: How do you do?
Fine, how are you?
How you come on?
Pretty good, sure as you're born.
Uncle Remus:
Stop jumpin', Br'er Rabbit,
you'll run out of breath.
Why don't you sit down an' calm yourself?
Brer Rabbit:
Well the grasshopper jump,
and so do the flea.
I do what I like, and I suits me!
[Repeat Chorus]
Uncle Remus:
The weather's good, the fishin's fine.
Now what do you do with all your time?
Brer Rabbit:
Oh, I zigs and I zags, I to's and I fro's.
That's what you're askin'
and that's what you knows.
[Repeat Chorus]
Brer Frog:
Mind now, Br'er Rabbit, better mend your ways.
You's headin' for trouble one of these days.
Uncle Remus:
Warnin' that rabbit is wastin' your breath!
Brer Rabbit:
Don't worry about me;
I can take care of myself!
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In the film, this takes place on the second of three animated sequences. Brer Rabbit is hopping along a road and singing "How do You Do?" Brer Frog and Uncle Remus join in, Remus interested in Brer Rabbit's lifestyle and Brer Frog concerned about what Brer Rabbit's doing living a dangerous lifestyle. It sets up to the point where Brer Rabbit asks a Tar Baby (set up by those sly Brer Fox and Brer Brer) "How do you do?" and the tar, of course, doesn't respond. Brer Rabbit starts to beat it up and ultimately gets trapped. (He gets out by being thrown in the briar patch, just like in the attraction).
In the attraction, Brer Rabbit, along with a few other birds outside, are first heard hearing the song. Then, geese and frogs are the first characters to sing "How do you Do?" inside the show building, setting up a friendly atmosphere of Southern courtesy. Next is Mr. Bluebird, who takes the role of Uncle Remus in this case, asking Brer Rabbit, "Now what do you do with all of your time?" And in case of the ride, the lyrics were changed for Brer Rabbit who now sings about leaving home (actually the first of the three segments of the cartoon of Song of the South). Mr. Porcupine (originally a character from America Sings at Disneyland) takes the role of Brer Frog as the warner: "He's heading for trouble one of these days!"
Still, Brer Frog takes the role of Remus for most of the attraction as the storyteller.
Photos:
Here is a link to the original sheet music:
http://www.songofthesouth.net/memorabilia/sheetmusic/us/how.html