Chairman Barry Jenkins announced the new appointment.
Clapp comes from the Department for Culture, Media and Sport, where he had been deputy director for the creative industries.
Clapp said: 'Cinema has a unique place in the culture of the nation. However, many overlook the economic contribution the sector makes, both nationally and at the community level. I look forward to working with CEA members to continue to give the UK cinema sector a strong voice, helping the sector address the challenges and opportunities ahead.'