At first I thought you might be looking for CUCURRUCUCÚ PALOMA. It was written by Tomás Méndez, a Mexican, in 1954, and has become very popular among Spanish-speaking singers. There are numerous recordings at YouTube. Americans who have recorded it include Joan Baez, Harry Belafonte, and Perry Como.

Then I found there is a song called COCORICO that has been recorded by some genuine Mariachis. It was written by Lorenzo Barcelata and recorded by Pancho Cataneo y sus Mariachis. I have found a few sound samples, but I am unable to transcribe them.