The BBC has already begun the search for Hall's successor.
The announcement follows weeks of speculation over who will run the London venue, home to both the Royal Opera and the Royal Ballet.
Mr Hall said he was taking the job at the beginning of April, as it was "too tempting an opportunity to turn down".
The House's former executive director, Michael Kaiser, left in December to become president of the John F Kennedy Center for Performing Arts in Washington.
The BBC has already begun the search for Hall's successor.
Internal candidates are expected to include Richard Sambrook, the deputy director of BBC news; Mark Byford, director of BBC World Service; Roger Mosey, the head of television news and Mark Damazer, the assistant director of news.
Outsiders whose names have been linked with the job include Nick Pollard, the editor of Sky News; Richard Tait, the editor-in-chief of ITN; and Steve Anderson, ITV's controller of news and current affairs.
Mr Hall's appointment was welcomed by Shadow Culture Secretary Peter Ainsworth.
He said: "He will take up his post at a challenging time and I wish him well in building on the achievements of Michael Kaiser."
The House's chairman, Sir Colin Southgate, said he was delighted that Mr Hall was joining the Royal Opera House.
"His extensive experience of managing complex budgets and working in a fast moving, creative environment make him ideally suited to this position."
Mr Hall said that as "a lifelong devotee" of the House, "it means a huge amount to me", and that the job would be a great challenge.
And BBC director general Greg Dyke paid tribute to his "remarkable career" at the corporation.
'Openness and accessibility'
"During his 10 years in the role of director of news he had transformed the BBC's news services," he said.
"In the year we have worked together I have been tremendously impressed both by his personal commitment and by the talented team he has built around him."
But writers and critics from the opera world were less enthusiastic about Mr Hall, saying he was seen to have little or no experience in opera or theatre management.
Mr Hall will be responsible for co-ordinating the numerous activities of the ROH with "special focus on the formulation of the annual budget", Sir Colin said.
He added that Mr Hall's appointment completed the "triumvirate newly appointed to lead the Royal Opera House in this new era of its history".
Antonio Pappano will take up his post as music director in September 2002, while Sir Anthony Dowell hands over to Ross Stretton at the end of the current season after 15 years as Director of The Royal Ballet.
Sir Colin said Mr Hall was well placed to understand the new challenges still facing the ROH, "in which art and education must take priority, with a focus on openness and accessibility".
Mr Hall worked his way up through the BBC's hierarchy after cutting his teeth in the corporation's newsroom in Belfast in the 1970s.
Controversy
After a spell as editor of the Nine O'Clock News he became head of the BBC's news and current affairs directorate in 1990.
Over the next 10 years he helped launch BBC Radio 5Live, BBC News 24 and BBC News Online, and oversaw the controversial move of the Nine O'Clock News to 10 o'clock.
But his application to become director general was turned down by the corporation's governors in 1999, with Greg Dyke taking the top job.
Although Mr Hall is leaving his job in news, his new position will no doubt hit headlines - the Royal Opera House has been the subject of controversy in recent years.
It reopened in December 1999 after a £241m refit - with £78.5m coming from public funds.
Culture Secretary Chris Smith demanded the venue's productions be accessible to all opera and dance lovers, well-off and not so well-off.