The BBC has apologised for revealing the Queen raised concerns with the government about why radical cleric Abu Hamza al-Masri had not been arrested.
The apology comes after security correspondent Frank Gardner told BBC Radio 4 of a private conversation he had with the Queen some years ago.
The BBC said it and Gardner were sorry for the "breach of confidence", which both "deeply regret".
On Monday, Abu Hamza lost his latest appeal against extradition to the US.
The European Court of Human Rights ruled the extradition could go ahead. The Home Office hopes this can be achieved within three weeks.
The Strasbourg court's decision means that the cleric and four other terrorism suspects can face terrorism trials in the US after delays going back to the late 1990s. In the case of Abu Hamza, he was first arrested in 2004.
The development was being discussed on Radio 4's Today programme on Tuesday morning when Gardner revealed details of his conversation with the Queen on the matter.
He said the monarch had told him, in a private meeting, how she had been upset that Abu Hamza could not be arrested.
The radical cleric had risen to prominence for his sermons in and around Finsbury Park mosque, which gained wide media attention for their content.
Gardner said the Queen had told him she had spoken to a former home secretary about the case.
In a statement, the BBC said: "This morning on the Today programme our correspondent Frank Gardner revealed details of a private conversation which took place some years ago with the Queen.
"The conversation should have remained private and the BBC and Frank deeply regret this breach of confidence. It was wholly inappropriate. Frank is extremely sorry for the embarrassment caused and has apologised to the Palace."
A spokeswoman for Buckingham Palace said it would "never comment on private conversations involving any member of the Royal Family".
The Home Office also said it would not comment on such conversations.
Former Home Secretary David Blunkett said "categorically" that the Queen never raised the issue of Abu Hamza with him.
"Not surprisingly," he said, "because my views and attitude in relation to this individual were very well known."
Abu Hamza and four other men accused of terrorism offences had fought against extradition for years, arguing at the European Court of Human Rights that they faced inhumane conditions in the US.
Abu Hamza is wanted over allegations he plotted to set up a terrorist training camp in the US and was involved in kidnapping Western hostages in Yemen. If convicted, he faces life imprisonment.
The case of Babar Ahmad - who, with co-accused Syed Talha Ahsan, is alleged to have run a jihadist website in London that provided support to terrorists - relates to a website run from London which, the US says, supported terrorism overseas.
Supporters of the pair say they should have been prosecuted years ago in the UK because the alleged crimes were committed in London.
Earlier this month, a businessman began the process of launching a private prosecution, saying that British suspects should be tried in the UK, not abroad.
Karl Watkin said: "The principle is simple - if you are British, and alleged to have done something criminal in this country, then you get prosecuted in this country.
"That's how the public interest is served. Contrary to reports, my motivation for prosecuting these two men in Britain is to establish this principle."
Mr Ahmad's father, Ashfaq Ahmad, added: "Because any crimes he is supposed to have committed are in this country, Babar should be tried in a British court and he should get a chance to prove his innocence in front of our courts here."