Kim Kardashian has given birth to a baby girl

Source close to KK tell TMZ ... it was a NATURAL CHILD BIRTH. She started having contractions Friday night and went into full labor early Saturday AM.

Our sources tell us the baby and Kim are both in good health. We're told Kim delivered the baby at Cedars-Sinai Medical Center this morning ... 5 weeks early.

Our sources say BF Kanye West was there for the entire thing -- in the delivery room by Kim's side. We're told Kanye cancelled his appearance at his own record release party last night when Kim told him she was feeling "a little off."

It is the first child for Kim and Kanye.