The former reality star's Lisa gave birth to their second child, a baby girl called Andy Rose Osbourne, on Saturday, 14 June.
They are already parents to daughter Pearl, three.
Jack confirmed the happy news by sharing a photograph to Instagram showing a bottle of alcohol-free champagne. His caption read: "This is how you celebrate when Mum is breastfeeding and Dad is sober. Thank you all for the kind words. #newbaby."
The new baby's arrival means Jack's parents Sharon and Ozzy Osbourne are now grandparents for the second time!
Lisa's pregnancy was announced via Instagram in February. The couple broke the news by sharing an image of little Pearl resting against her mum's baby bump.
A thrilled Lisa said: "I'm pregnant! Pearl is very happy to become a big sister. She diligently reads her 'I'm a big sister' books every night & recites to me her plans of bathing, feeding, singing to, and holding the baby. We are more than halfway through this thing & are very excited to be parents again! Thanks for all the well wishes."
Jack and Lisa have been married since 2012.
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