The album is Stanley Holloway, "The Elephant Alphabet: and Other Songs and Rhymes for Children" (Riverside Wonderland RLP 1415).
The album's front cover is viewable if you click "cover" on that page.
If you click on "mp3s" you can download a zipped folder which contains 2 mp3 files: one, I presume, for each side of the LP. There is no track list, so I can't confirm the song title, but the song in question begins at 10:35 on the first file (Side 1?).
Here's my transcription:
THE GIRAFFE IS LAUGHING / THE LAUGHING GIRAFFE POLKA (possible titles)
Do, do, do, do, do you know what happened at the zoo? What happened at the zoo? What are things coming to? Do, do, do, do, do you know what happened at the zoo? It really, really, really has come true.
The giraffe is laughing. Yes, indeed, he's laughing. Yesterday he started laughing. Now he's laughing more and more.
The giraffe is laughing. Can't you hear him laughing? Strange indeed he should be laughing! He never made a sound before.
Said the camel to the bison: "This is really quite surprisin'." Said the leopard to the llama: "He'll stop laughing when he's calmah." Said the lion to the lynx: "This is very queer, methinks." Said the zebra to the rabbit: "Looks like it's become a habit."
Do, do, do, do, do you know what happened at the zoo? It happened at the zoo. It happened at the zoo. Do, do, do, do, do you know what happened at the zoo? You laughed at him so now he laughs at you.
The giraffe is laughing. Can't you hear him laughing? Strange indeed he should be laughing! He never made a sound before.
[STARTING OFF RHYTHMICALLY, MUSICALLY:] Ah, ha-ha! Ah, ha-ha! Ha-ha-ha-ha-ha! [DEGENERATING INTO UNCONTROLLED LAUGHTER:] Ha-ha-ha-ha-ha....