Cox is effectively swapping jobs with Moyles who currently hosts the drivetime slot. Her career trajectory is almost identical to that of Ball, who hosted the Radio 1 breakfast show and now hosts the drivetime show on guitar and new-music station Xfm.
Ball said that Cox had been her bridesmaid and the godmother to her child with husband Fat boy Slim, and she said she is her rival.
"I'm really chuffed for her. If she thinks she she's going to be taking any of my Xfm listeners she can think again. The gloves will be coming off and, believe me, I've got strong nails," Ball said.
The changes are set to take place in January, although no date has been set.
The flagship breakfast show was the worst hit of Radio 1's programmes in the last set of Rajar audience figures for the second quarter of 2004, losing 473,000 listeners, half of which were from London.
Cox said she had had a "great time" presenting the show, but was looking forward to a lie in. She has worked at the station since September 1999, when she co-hosted the Saturday lunchtime show alongside Emma B. She took over the breakfast slot from Ball in summer 2000.
Radio 1 controller Andy Parfitt said: "[Moyles] was my first and only choice as successor to Sara, who has done a great job over the last three-and-a-half years."
The change of presenters will pit Moyles against Johnny Vaughan, who was last week confirmed as a replacement for Chris Tarrant on Capital Radio, ending over a year of speculation about how long he would stay at the station.
Parfitt said: "In a market that is getting more competitive by the month, I think Chris is going to be just what our audience is looking for."
Radio 1 has been suffering with falling ratings, sliding below the 10m mark in the latest Rajar figures. The station reported an audience of 9.9m in the second quarter of 2003, down from 10.5m a year ago and 10.3m last quarter.
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