Kim Kardashian and Kanye West already revealed the name of their third child -- Chicago West -- but now we've got her birth certificate.

According to the doc -- obtained by TMZ -- Chicago was indeed born at 12:47 AM on January 15 and weighed in at 7 lbs. 6 oz.

The delivery doctor is listed as Dr. Paul Crane ... who also delivered North and Saint -- as well as just about every other Kardashian.

As we first reported, the birth was via surrogate.