The 47-year-old will replace Chris Evans, 52, on the UK’s most listened-to radio programme.
She said she was not “under-estimating the enormity of the task ahead”.
The Strictly Come Dancing: It Takes Two host said: “I’m absolutely thrilled to be following in the giant footsteps of Chris Evans as the host of the Radio 2 Breakfast Show.
“To be the first woman to present this very special show is both an honour and privilege.
“Believe me, I’m not under-estimating the enormity of the task ahead, to follow not one but two of my broadcasting idols into such a well-loved show is somewhat daunting.
“But I hope, in the same way that Chris made this show his own after taking over from the wonderful Sir Terry Wogan, that, with a top team alongside me, I can bring the fabulous Radio 2 audience a show they want to wake up to.”
Bob Shennan, director of BBC Radio and Music, said: “The Radio 2 Breakfast Show is arguably the most coveted job in UK radio with a long and illustrious history, and I’m thrilled that the baton is now passing to Zoe, who I know will make the show her very own.”
Ball, the first solo female Radio 1 breakfast show host, was favourite to take over the high-profile role on Radio 2.
She was at the forefront of the “ladette culture” that swept through broadcasting because of her hard-drinking party lifestyle.
She recently made a programme for Sport Relief following the loss of her partner, Billy Yates, who was found dead at his home at the age of 40.
Ball hosts a live Saturday afternoon show on Radio 2 and has stepped in for Evans on the breakfast show.
BBC boss Lord Hall recently said the disclosure of Evans’ pay packet, of at least £1.6 million, was one factor behind his decision to quit Radio 2.
He is moving to Virgin Radio, “my spiritual home”, almost 20 years after he left the station.
Evans previously played a trailer for a Fatboy Slim documentary on Radio 2, airing on Wednesday night, and said: “This family is taking over - more of that later.”
He told listeners: “Don’t think you’re going to be disappointed.”
Ball was previously married to DJ and record producer Fatboy Slim, whose real name is Norman Cook.
Evans took over from Sir Terry Wogan as presenter of the show in January 2010.
Sara Cox will continue to cover the Radio 2 Breakfast Show for 10 weeks each year, while Ball is away.