Cover Story

Kim Kardashian: Inside Her Dramatic Delivery

Update Wednesday June 19, 2013 11:45 AM EDT Originally posted Wednesday June 19, 2013 08:00 AM EDT

Leave it to a Kardashian girl to make a dramatic entrance!

That's exactly what happened when Kim Kardashian, 32, delivered a daughter roughly three weeks early.

With exclusive interviews with members of the couple's innermost circle, this week's PEOPLE cover story looks inside the emotional, harrowing night Kardashian and her boyfriend of 16 months, Kanye West, 36, spent at Cedars-Sinai Medical Center in Los Angeles awaiting the birth of their firstborn.

Admitted June 14 due to complications, Kardashian – whom sources confirm was about 37 weeks pregnant – went into premature labor.

West, meanwhile, who'd returned from promoting his new album Yeezus in Europe just hours before, rushed to her side.

By the next day, Kardashian had given birth to a girl, "the prettiest, tiniest thing with a full head of straight, dark hair, a cute little nose and Kim's lips," says one of several close friends who visited her over the weekend. "She's already smiling!"

Though she's been spending time in an incubator to regulate her body temperature, a standard course of treatment for premature babies, Baby K – whose parents have nearly decided on a name – is "doing very well," says a hospital source.

As for West and Kardashian, who'd been dogged by rumors about their relationship in the months leading up to the birth, the new parents are over the moon.

For more details on Kardashian's emotional emergency delivery, their new baby, and whether a wedding is on the way, pick up the new issue of PEOPLE, on newsstands Friday