The appointments were made by Jenny Abramsky, the director of radio and music, and Andy Parfitt, the controller of Radio 1 and 1Xtra.
Parfitt has handed over responsibility for the stations' output because he has recently been tasked with making BBC content in all media more relevant to 12- to 16-year-olds by Mark Thompson, BBC director-general.
Cooper was previously head of mainstream at Radio 1 while Clarke will retain her current role of festival director of the BBC's Electric Proms.
A third change gives Ian Parkinson, formerly head of specialist music and speech at Radio 1, the role of head of editorial standards for Radio 1 and Radio 1Xtra and music development projects for radio and music interactive.
Parfitt said: "Ian has an extraordinary range and depth of experience that will mean that we can continue to push boundaries to reach our young audience while maintaining the highest editorial standards.
"This restructure is essential to futureproof both networks and ensure that the strategies for Radio 1 and 1Xtra are delivered effectively."
Radio 1's share of total radio listening in the third quarter was 9.8%, compared with 9.4% a year earlier, while 1Xtra's was 0.2%, compared with 0.3%.
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