If Gwyneth Paltrow and Coldplay’s Chris Martin seem a bit giddy these days, you’ll have to forgive them: Not only are they expecting a baby, but they’re also newlyweds, a source close to the actress tells PEOPLE.

Confirming recent wedding rumors, the source says Paltrow, 31, and Martin, 26, tied the knot in Santa Barbara, Calif., on Dec. 5, soon after receiving their marriage license at the Santa Barbara County courthouse, the magazine reports in its upcoming issue.

The surprise nuptials — no family members, including Paltrow’s mom, Blythe Danner, attended the ceremony — capped a whirlwind week for the pair, who had looked “very intimate” while dining Nov. 29 at New York City’s Da Silvano restaurant, says general manager Alessandro Bandini. “I saw her look at Chris and blush. They acted like a young couple in love.”

On Dec. 1, after visiting a doctor’s office on Manhattan’s Park Avenue, they acted like any other young, happy, expectant couple, with Martin playfully snuggling and patting Paltrow’s tummy. (The photos also appear in the upcoming PEOPLE.)

After their baby announcement on Dec. 3, best wishes immediately began pouring in. When Paltrow taped an appearance on “The Ellen DeGeneres Show” Dec. 4, the comedian surprised her with baby gifts, including a $3,000 Silver Cross Balmoral navy blue handmade pram (a British baby carriage). The baby is due “summer-ish,” Paltrow told DeGeneres. “It’s exciting.” However, the actress admitted to Jay Leno, “The Jewish part of me is superstitious about talking about it, so when people say, ‘Are you going to find out about the sex?’ ‘Do you have weird food cravings?’ or ‘How does your mom feel?’ I don’t want to answer those questions.” (Okay, but how does Mom feel? “I spoke to Blythe and she was thrilled,” says one source close to Paltrow. “She’s been wanting grandkids forever.”)

Though Paltrow and Martin — who divide their time between London and Manhattan — have been spotted house-hunting in the Belsize Park area of north London, they had been cagey whenever reporters inquired about an engagement. Now they’re keeping quiet about the marriage. After the hush-hush trip to Santa Barbara, the couple escaped to the posh Esperanza Resort outside Cabo San Lucas, Mexico.

For more on Gwyneth and Chris’s whirlwind wedding, pick up the latest issue of PEOPLE, on newsstands Friday.