The actress took to Instagram on Wednesday to reveal the happy news, posting a series of pink pictures starting with the date of her baby's birth.
The little girl – who they have named Rani Rose Hudson Fujikawa – was born on Tuesday, and Kate wrote in another picture: "We have decided to name our daughter Rani (pronounced Ronnie) after her grandfather Ron Fujikawa.
"Ron was the most special man who we all miss dearly. To name her after him is an honor."
Another picture read: "Everyone is doing well and happy as can be. Our family thanks you for all the love and blessings that have been sent our way and we send ours right back."
Kate, 39, captioned the pictures with simply: "She's here".
Kate already has two sons from previous relationships; Bingham Dawn, seven, with ex-fiance Matt Bellamy, and 14-year-old Ryder Russell from her marriage to Chris Robinson.
Last month Kate showed off her huge baby bump in an Instagram mirror selfie.
Wearing an open caftan, the Hollywood star looked positively glowing as her belly protruded out in front of her.
And she captioned the moment with the simple word: “Outie” – showing her belly button was sticking outwards.
Kate previously spoke about her relationship with Danny, revealing: “We’ve known each other for 15 years. He’s my best friend’s stepbrother.
“We kinda skipped through the part of knowing what the hiccups were. You kinda skip the part of going, ‘Oh, you’re like that?'”
“In terms of family, I’ve known his family forever. To have that connection is really powerful."
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