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's difficult to deal with a highly competitive colleague. Every time you submit work, they try to find ways to outperform you. Although you have no interest in pitting your talent against theirs, this other person thinks very differently. Ignoring them won't be possible. A good way to disarm this pest is to invite them out for coffee. Describe what you've observed and ask for a truce. Irrational people are often thrown off balance when their behaviour is addressed in a direct manner.More