The British National Party won two MEPs in June's European elections
British National Party leader Nick Griffin will be allowed to appear on Question Time, the BBC Trust has said.
The BBC's governing body had convened a meeting to consider complaints from Welsh Secretary Peter Hain and others about Mr Griffin's appearance.
The trust said it was "a question of editorial judgment" whether it was appropriate for the BNP to appear.
But Mr Hain insisted the BBC had "made one of the biggest mistakes in its proud history".
In an article for Thursday's Guardian newspaper, BBC director general Mark Thompson has said the case against inviting the BNP to appear on Question Time is "a case for censorship".
The complaints had been forwarded to the trust on appeal after BBC management rejected them.
A special committee, chaired by former Newsnight editor Richard Tait, issued a statement following its ruling on Wednesday evening.
It said: "We have concluded that the decision as to whether it is appropriate in all the circumstances for the BBC to allow the BNP to participate on Question Time is a question of editorial judgment, which the charter and framework agreement reserve for the director general."
The committee said concerns about the potential content of the show could not be addressed before it was filmed.
"In circumstances where the programme is pre-recorded and edited before transmission it is premature and hypothetical to rule on these criticisms in advance of broadcast," it said.
"We have written to the director general asking him to engage personally to ensure that the broadcast programme complies with the general law, the BBC's editorial guidelines and all other regulatory requirements."
The statement said the committee was "aware of the debate and public controversy on this issue and that this is a matter of considerable importance to many licence fee payers".
"Given the trust may be asked in future to determine appeals arising out of the broadcast of this edition of Question Time it would be inappropriate for us to make any further comment," it added.
'Gift of credibility'
Mr Hain said he regretted that the BNP leader would be able to appear.
"I am very sad that the BBC I grew up with and held in such high regard has made one of the biggest mistakes in its proud history," he said.
"Nick Griffin has today confirmed that I was right: he feels that the BBC have given him an early Christmas present.
"This gift of credibility will last him a political lifetime."
The Welsh secretary has argued that the BNP was "an unlawful body" after a court ruling on its membership policy.
Mr Griffin agreed last week to use "all reasonable endeavours" to revise its constitution so it did not discriminate.
The BBC's director general Mark Thompson said the court case did not "legally inhibit" the BBC from allowing Mr Griffin on the programme, but Mr Hain disagreed.
"Why does the BBC not suspend their invitation to Question Time, at least until they have changed their constitution to satisfy the court?" he said.
"Never in 30 years has the BBC invited a convicted criminal found guilty of inciting racial hatred on to this flagship programme."
In a letter to Mr Hain, Mr Thompson said the BBC had been advised that if there were to be any election - local or national - tomorrow, the BNP would still be able to field candidates.
Nick Griffin faced legal action over the BNP constitution
In the Guardian article Mr Thompson said arguing against the invitation to appear was making "a case for censorship".
"The case, in other words, that (in the opinion of those who make it) the BNP's policies are so abhorrent and so liable to sow hatred and division that they should be excluded from this form of public discourse altogether," he said.
He added that where organisations were deemed to be "beyond the pale" they were proscribed and/or banned from the airwaves by - and only by - governments.
Question Time "carefully" studied the support gained in elections by each of the parties before deciding who to invite and how frequently, he added.
"Question Time is an opportunity for the British public to put questions to politicians of every ideological hue. Politicians from the UK's biggest parties appear most frequently, but from time to time representatives of parties with many fewer supporters... also take their seats on the stage," he said.
"It is for that reason - not for some misguided desire to be controversial, but for that reason alone - that the invitation has been extended."
The BBC's chief political adviser, Ric Bailey, said the decision to have Mr Griffin on Question Time was based on the party's success in June's European elections, at which it won more than 940,000 votes and two seats, he said.
"We absolutely stand by that judgment, even though there's obviously been a lot of controversy about it," he said.
Anti-fascist campaigners have said they will stage a demonstration against Mr Griffin's appearance on the show outside the BBC's Television Centre, in west London, on Thursday.
As well as Mr Griffin and Mr Straw, panellists are expected to include Conservative peer Baroness Warsi, Lib Dem home affairs spokesman Chris Huhne and playwright and critic Bonnie Greer.