The comedian and model announced they were expecting a child in a social media post in May after previously opening up on their devastating miscarriage.
Following the birth of their baby, Jack shared two adorable photos of the new parents, one showing him smiling at the camera while cradling his daughter. The other saw the pair beaming while Roxy cuddled the baby in a hospital bed.
Sharing the post with his followers, Jack wrote: "Well this just happened! Utterly overwhelming and joyous in ways I couldn't have even imagined."
He went on to praise Roxy, saying she had been "amazing" through the pregnancy and birth and is already the "greatest mum ever".
He continued: "In awe of my partner @roxyhorner who has been amazing throughout this journey and is going to be the greatest mum ever. So excited to have started a family of my own.
"Having vowed I'd never be that guy I am now 100% going to be the parent that shows everyone endless pictures of their kid. I also wore Sketchers to hospital, I think I must accept I am a full-blown dad now."
However, he didn't share the little one's name in the post, saying "after this weekend I think 'rice, rice, baby' has a nice ring to it."
Roxy also shared a picture of their baby to her Instagram page, showing her in a white, lace dress nuzzling into her chest. She simply captioned the post: "The love of my life [heart emoji" with the pair's celebrity friends flocking to the comments to congratulate them.
Pixie Lott wrote: "Soooo beautiful, so happy for you both!!!!", while Ferne McCann said: "Congratulations darling. Welcome to the pink club".
Football icon Jamie Redknapp, models Belle Lucia and Lorena Rae, and comedian CJ Franco also shared their congratulations with the couple.
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