A spokesperson for the couple confirmed the news to The Sun, saying, 'Yes, I can confirm it's true.'
Eddie and Hannah have chosen to name the new arrival Iris Mary, with Iris meaning 'rainbow' in Greek, and Mary meaning 'wished-for child' in Hebrew - aww!
The pair confirmed the pregnancy back in December, revealing that they had chosen not to find out the sex of their child in advance, and Eddie later opened up about his impending role as a new father in an interview in March.
'It's our first child, so obviously there's a mixture of emotions,' he confessed. 'Joy, excitement and a bit of fear. I can't wait. I'll be a hands-on dad.'
He added that he couldn't believe how lucky he'd been, both career and family-wise, in recent years. 'I still pinch myself — every actor thinks about living the Hollywood dream and making it over here is incredible.'
'I'll always stay grounded and I don't take anything for granted.'
Eddie and Hannah pictured in March 2016
The Oscar-winner met his now-wife while attending Eton College, when she was enrolled in a nearby boarding school for girls. They remained friends for over a decade, and Hannah eventually became his publicist, with romance blooming on a trip to Florence in 2012, when he was shooting Les Miserables.
They married in a romantic Christmas ceremony in 2014.
'It was the most wonderful thing,' Eddie said at the time. 'We had Christmas off and so we had a Christmas wedding. It had been a few months of travelling promoting the Theory of Everything. It was a small wedding, but it was wonderful.'
He'll now be juggling taking care of Iris with promoting his upcoming Harry Potter spin-off, Fantastic Beasts and Where to Find Them, which hits screens in November this year.