Wide-eyed and beautiful – that’s rock star Pink’s brand new baby girl Willow Sage!

Dressed all-in-black, proud parents Pink (Alicia Beth Moore) and her husband Carey Hart were all smiles as they carried their month-old daughter around in a baby basket in Malibu, California on Wednesday.

Pink and her motocross hubby Carey Hart are proud parents to a beautiful baby girl.

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The singer gave birth to the couple's first child Thursday -- they have named her Willow Sage Hart.

RadarOnline.com was first to report that Pink had arrived at Cedars-Sinai hospital around 12.30pm Thursday afternoon.

Pink was scheduled for a C-section as the baby was breached and a source tells us the operation went without a hitch and both mother and baby are doing well.

"Everything went exactly to plan and Pink and Carey are absolutely thrilled," the source says.