Channel 4 aired the Dispatches show in January 2007
West Midlands Police and the Crown Prosecution Service have apologised for accusing the makers of a Channel 4 documentary of distortion.
The apology and the promise of £100,000 were made at the High Court on Thursday.
It follows comments made about a Dispatches programme, Undercover Mosque, which tackled claims of Islamic extremism in the West Midlands.
The police statement said the force was wrong to make the allegations.
A press release issued by the police and the CPS in August 2007 claimed the Dispatches programme, broadcast in January of that year, misrepresented the views of Muslim preachers and clerics with misleading editing.
'Damage and distress'
One preacher was shown saying a homosexual should be thrown off a mountain, another that women were born deficient.
Footage from the Channel 4 programme
Police also reported Channel 4 to television watchdog Ofcom for "heavily editing" the words of Islamic imams.
But in November, Ofcom rejected the police and CPS claims, and Channel 4 said it was going to sue the CPS and police for libel.
The statement, released to the media after the High Court hearing by West Midlands Police, said they accepted there had been no evidence that Channel 4 or the documentary makers had "misled the audience or that the programme was likely to encourage or incite criminal activity".
It added that the Ofcom report showed the documentary had "accurately represented the material it had gathered and dealt with the subject matter responsibly and in context".
The police statement concluded: "We accept, without reservation, the conclusions of Ofcom and apologise to the programme makers for the damage and distress caused by our original press release."
Bigotry and extremism
Kevin Sutcliffe, deputy head of current affairs at Channel 4, said the apology was a vindication of the programme team in exposing extreme views.
"Channel 4 was fully aware of the sensitivities surrounding the subject matter but recognised the programme's findings were clearly a matter of important public interest.
"The authorities should be doing all they can to encourage investigations like this, not attempting to publicly rubbish them for reasons they have never properly explained," he said.
David Henshaw, executive producer of Hardcash Productions, who made the documentary, said it was a thorough and detailed programme, made over nine months and at personal risk to the undercover reporter.
Channel 4 boss Julian Bellamy said they had had no choice but to pursue action when the police and CPS refused to withdraw their remarks.
The programme infiltrated a number of mosques, one of which was Green Lane Mosque in Small Heath, Birmingham.
An undercover reporter claimed to provide evidence that certain speakers preached messages of religious bigotry and extremism.
Police initially investigated whether three of the people shown in the programme could be prosecuted for inciting terrorism or racial hatred.
But they later switched their attention to the documentary makers, suggesting they may have been guilty of stirring up racial hatred.
Channel 4 said £50,000 would be donated to the Rory Peck Trust for freelance journalists and their families.
The broadcaster will also receive £50,000 to cover legal costs.
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