I'm a bit confused. I listened to the song again, and although numerous websites say the title of the song is COCORICO, it sounds like they're singing "coconito." So I tried Googling with "coconito" and I found this:

I found this image of the sheet-music cover. Most of the words on this cover are French, not Spanish.

Cocorico!
(Coconito)
Valse mexicaine [Mexican waltz]

Paroles espagnoles & musique de [Spanish words and music by] L. Barclata
Paroles françaises de [French words by] Chamfleury & Liogar
Éditions espagnoles [Spanish editions] Julio Garzon [I assume that's the publisher]
[There is some tiny print below that, but I can't read it. It might include the word "Paris".]

The most important thing about this is: it confirms that "coconito" might be a good thing to search for. So then I was able to find this:

From Cancionero Infantil Mexicano [Mexican Children's Songbook] by Antonio Avitia Hernández (México, D.F.: Selector, 1996.), page 32:


EL COCONITO
Lorenzo Barcelata

Coni, coni, coconito,
coni coni qué caray,
yo le daba su maicito,
antes sí, pero ahora no hay;
coni, coni, coconito,
coni, coni, qué caray,
yo le daba, su maicito,
siempre sí, pero ahora no hay.

En medio de la sabana,
gorgorea un coconito,
y todos los días su nana,
le baja maíz al cerrito,
así lleva p'su hermana,
a lo troje su maicito;
coni, coni, coconito,
coni, coni, coconito.

[Then the first verse is repeated.]

It also has musical notation for the melody line.

I listened to the recording again and it seems to match fairly well—although Pancho Cataneo y Sus Mariachis sing only the first verse.

The second part (which probably shouldn't be called a "verse") goes to a different melody. The only singer I found who sings this is Aparo Ochoa, on her albums "Cancionero Popular Mexicano" and "Por Siempre." I think she also sings some other words that I don't have.

If you want to search for recordings, it's better to search for "coconito" than "cocorico." Sometimes the title is given as COCORICO, sometimes COCONITO, and sometimes COCORICO (COCONITO). But beware: there are other unrelated songs called COCORICO.

I won't bother posting Google's translation. It was ungrammatical and didn't make sense.