Three weeks before her due date, amid swirling rumors about their relationship, Kim Kardashian, 32, gives birth to a baby girl – as , 36, rushes to her side. This week's issue of PEOPLE goes inside the real story behind Kim and Kanye's baby joy – and drama. Watching her deliver, "he was very emotional," says a source who works with West (one of several of the pair's confidantes who spoke to PEOPLE on the condition of anonymity to honor their request not to share many details of the birth). After a harrowing night, they welcomed a girl on June 15.

Because of her low weight, Baby K – whose name hadn't been revealed at press time but who sources said would have the first initial K, just like Mom and Dad (**UPDATE** they named the baby "North" according to the birth certificate) – was put in an incubator to regulate her body temperature, a standard treatment for preemies. While the pair kept close watch over their fragile newborn, the usually vocal Kardashian clan stayed mum. "This was a lot to go through," says pal Robin Antin, founder of music group GRL, who visited her in the hospital. "I'm sure she was exhausted, but she handled it like a pro." A day after the birth, mom Kris Jenner still seemed stunned, telling reporters at the Daytime Emmy Awards, "It all happened so fast!"

Despite the scare, by June 17 everything was "good. The baby is the prettiest, tiniest thing with a full head of straight, dark hair, a cute little nose and Kim's lips," says a close friend who saw Kardashian after the birth. Adds pal Brittny Gastineau, "She's beyond beautiful." As for the new mom, who was due the first week of July (not July 11, as had been reported), "she is a natural mother," says Antin. "Kanye is so happy. The energy between them is amazing."

By the time she was preparing to be discharged, the happy new mom had barely put her newborn down. "For premature babies with no major complications, it's very important that they are held close to the mother," says a source at the hospital. "It helps them gain weight faster." West, meanwhile – who "hasn't left her side," says the Kardashian friend who visited her – was the proudest papa in town. They'll start planning their next project, a wedding, sooner rather than later. "He definitely wants to marry her," says the Kanye source. "She doesn't want another over-the-top wedding," says the close friend. "She wants to be surrounded by people she loves."

More on Kim Kardashian and Kanye West's baby is featured in the new issue of PEOPLE, on newsstands everywhere now.

*Also In This Week's 'People'*
Katie Holmes A Year After The Split With Tom: "She's Not Looking Back"
Princess Kate: Inside Preparations For The Royal Baby
Miley Cyrus Copes With Her Parents's Divorce: "I'm More Concerned For My Sister"
Heather Locklear On Being "Cut Out Of" The First Wives Club Movie