Gibbs spent more than 20 years at the BBC, where he started out as a graduate news trainee and worked most recently as head of programmes for BBC World News.
He became director of programmes for Al Jazeera International in 2002 as the channel prepared for launch and also worked for Channel Five, Discovery, Channel Four and Carlton TV during his career.
Fran Unsworth, deputy director of news and current affairs at the BBC, said: “Paul fought a long battle in recent years with cancer, which he bore with his characteristic good humour and optimism.
“He was married to Eileen Fitt who, many will know, was an output editor in BBC News on Newsnight and the Nine, then Ten. Our thoughts are with her and their children.”