The BBC has apologised over reports claiming millions of pounds raised by Band Aid was used to buy arms.
In March, World Service's Assignment said cash raised by charities to help Ethiopia had been diverted by rebels.
The BBC has admitted that Assignment gave the impression that Band Aid and Live Aid money had been diverted despite no evidence to back that up.
It apologised for further TV, radio and online reports which actually stated that Band Aid money had paid for arms.
The BBC's Editorial Complaints Unit found in its ruling that there was no evidence to support such statements and that "they should not have been broadcast".
"The BBC wishes to apologise unreservedly to the Band Aid Trust for the misleading and unfair impression which was created," it added.
The original investigation by BBC World Service Africa editor Martin Plaut included claims that substantial amounts of aid from western governments and charities went into rebel-held areas of Tigray province in 1985 and was used to buy weapons.
Former BBC chairman Michael Grade, a trustee of the Band Aid Trust, said Assignment had "sexed up" its story by "trying to smear Live Aid through this programme through the use of all the music from Live Aid and using Bob Geldof's name".
He added: "We're very glad finally to be able to reassure all the millions and millions of people around the world over 20-odd years who've given millions of pounds to Band Aid and Live Aid to relieve suffering that, of course, the money did not go to arms."
He said the BBC had made "a terrible, terrible mistake, they've damaged 24 years of work, they've damaged the public perception of giving aid to relieve starving people around the world".
And he questioned why it had taken seven months since a complaint by the Band Aid Trust for the BBC to make an apology.
Director of global news, Peter Horrocks, told Today the examination of the complaint "was a detailed one - there were lots of detailed aspects that have been gone into and fresh research was done".
"The thoroughness of the complaints procedure and the fact that the BBC can wholeheartedly acknowledge that errors have been made I would regard as being a strength," he added.
He said the BBC accepted that an unfair impression had been created by the production of Assignment "including, for instance, the use of Band Aid music".
"What we should have specifically said was there was no specific evidence and we're apologising today to the Band Aid Trust and we're also apologising personally to Bob Geldof."
The BBC admitted that further reports on outlets including Radio 4 and the BBC News website went further "than the programme itself in stating that millions of pounds raised by Band Aid and Live Aid had been diverted to buy arms".
It also apologised to Bob Geldof - the driving force behind Band Aid and Live Aid - for implying he declined to be interviewed at the time "because he thought the subject too sensitive to be discussed openly".
Mr Geldof said: "This was an unusual lapse in standards by the broadcaster and, most critically, the World Service.
"It was Michael Buerk's frontline reports for the BBC from Ethiopia which prompted me to act and establish Band Aid in the first place and I recognise the important journalistic and humanitarian role the BBC has played in our story."
He welcomed the apology and said he hoped it would "begin to repair some of the appalling damage done".
On-air apologies are due to be made on all BBC outlets that broadcast the claims.