The 5Live presenter was accused of mocking the duchess's racial heritage.
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Baker said: "Just got fired from @bbc5live."
A BBC spokesperson said: "This was a serious error of judgment and goes against the values we as a station aim to embody. Danny's a brilliant broadcaster but will no longer be presenting a weekly show with us."
Harry and Meghan, whose mother Doria Ragland is African American, their new son was named Archie Harrison Mountbatten-Windsor.
After an initial backlash on social media, Baker said: "Sorry my gag pic of the little fella in the posh outfit has whipped some up. Never occurred to me because, well, mind not diseased.
"Soon as those good enough to point out it's possible connotations got in touch, down it came. And that's it."
The 61-year-old presented a weekend show on 5Live.
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