Rossdale tweeted news of the birth and said Bowie and Flynn were their mothers' maiden names.
Apollo's brothers are five-year-old Zuma and Kingston, seven.
Rossdale, 48, is best known as the lead singer of the rock band Bush and Stefani, 44, is the lead vocalist of the band No Doubt, and has released two hit solo albums. She is also a fashion designer.
The couple met in the mid-1990s and were married in 2002.
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It feels like friends are few and far between. If you're not getting the help you need, ask why. It may be because you have been so preoccupied with work. Skipping special occasions and failing to return phone calls causes resentment. You have to be more engaged to sustain loving bonds. Call someone you've missed. It may take a few tries, but it's worth the effort. Be forthright about why you have been out of touch. Communication is the cornerstone of every healthy relationship.More