Cox, 24, who presents Channel 4's Big Breakfast - another seat once filled by Ball - said she was "thrilled to bits to be taking over from one of my best mates".
The Pretty Polly girl, who once admitted to "being so green I would have strapped a chicken to my head if they asked me", will have her work cut out keeping the show's 6.9m listeners. After a rocky start, Ball drew hundreds of thousands of new listeners, eclipsing her old mentor Chris Evans, who hosts a rival show on Virgin.
Cox's appointment ends months of speculation about a successor. Chris Moyles, who presents the station's afternoon show, the comedians Ant and Dec and the Big Breakfast's Johnny Vaughan were all in the running at one stage.
Ball is leaving to start a family with her husband, Norman Cook, the dance music star Fat Boy Slim, in Brighton and to concentrate on TV work.
Cox, who is to marry Leeroy Thornhill from The Prodigy, said her salutary experiences on the Girlie Show had taught her when to say no. Even so, she grabbed this job with both hands. "I know it's going to be hard; Zoe knows that better than anyone. She will be the first person I ring if I have any problems. She has done the job brilliantly, and she has had my best interests at heart."
Cox will take over in April but will stop presenting her Saturday Radio 1 show with Emma B and the Sunday Surgery show immediately.
The Radio 1 controller, Andy Parfitt, said Cox would take the show in a new direction. "It is not a replacement for Zoe but a whole new show, and it is going to set the agenda for each day on Radio 1."
