The Law Lords have ruled that radical cleric Abu Qatada can be deported from the UK to Jordan where he faces jail for terrorism.
Abu Qatada, 48, one of Europe's most influential extremists, had alleged that his conviction in Jordan was based on evidence extracted by torture.
The home secretary said she was "delighted" at the decision, calling him "a truly dangerous individual".
Abu Qatada's lawyer said she had lodged an application to appeal.
Gareth Peirce has submitted an application to the European Court of Human Rights. Her client cannot be deported until the appeal bid has been considered.
The five Law Lords who unanimously backed Abu Qatada's removal also supported the deportation of two Algerian terrorism suspects, known as RB and U, whose cases covered similar grounds.
Last year the Court of Appeal blocked Abu Qatada's deportation after accepting his argument that he had not faced a fair trial in his absence.
He was released on bail - but then re-arrested and returned to Belmarsh high security prison because security officials said they had intelligence that he was considering fleeing the UK.
He was first arrested in the wake of the 9/11 attacks amid allegations that he was one of the most influential Islamist preachers in Europe, playing a critical ideological role. One judge described him as Osama bin Laden's right-hand man in Europe.
His convictions in Jordan relate to an alleged conspiracy to bomb hotels in the capital Amman along with allegedly providing finance and advice for other plots.
The two Algerian men, who also lost their cases, had alleged that they faced torture if returned to their home country. They have been held in jail pending deportation because the Home Office said they posed a serious threat to national security.
Ms Smith said the ruling "highlights the threat these individuals pose to our nation's security".
"I'm delighted with the Lords' decision, a decision that agrees with us that we can remove Abu Qatada, a truly dangerous individual, from the UK," she said.
"I have now signed a deportation order which will be served on him later today. My priority is the safety of this country and I want him removed as quickly as possible."
The home secretary said the judgement would "bring other deportations a step closer".
Tory leader David Cameron told the BBC he supported Abu Qatada's removal from Britain.
"We should be a safe haven for people fleeing persecution, but we should not be a soft touch," he said.
"People who come here and radicalise young men and encourage them to do these dreadful acts, it's just unacceptable and they should be deported."
The UK has signed a string of diplomatic agreements with Middle Eastern and African countries designed to guarantee fair treatment of anyone deported from the UK on grounds of national security.
Critics have described these deals as legally worthless - Tom Porteous, director of Human Rights Watch, told the BBC they were "flimsy and unenforceable".
The BBC's security correspondent Frank Gardner said the Jordanians were very keen to have Abu Qatada back, but some there were concerned that his fate could become a complicated political issue which until now the country has avoided dealing with.
Lord Phillips of Worth Matravers, sitting with Lords Hoffmann, Hope, Brown and Mance, said that while evidence used against Abu Qatada may have been extracted by torturing another suspect, the issue for the British courts was whether the cleric could get a fair trial in Jordan, irrespective of how the evidence had been obtained.
Lord Phillips said: "The prohibition on receiving evidence obtained by torture is not primarily because such evidence is unreliable or because the reception of the evidence will make the trial unfair. Rather it is because 'the state must stand firm against the conduct that has produced the evidence'.
"That principle applies to the state in which an attempt is made to adduce such evidence. It does not require this state, the United Kingdom, to retain in this country, to the detriment of national security, a terrorist suspect. What is relevant is the degree of risk that Mr Othman [Abu Qatada] will suffer a flagrant denial of justice if he is deported to Jordan."
In the case of the other men, the lords said, the Algerian Ministry of Justice has provided the UK with a written guarantee of fair treatment and trial, including a pledge to respect their "human dignity .. under all circumstances".
Human rights group Amnesty International said it was "gravely concerned" about the ruling's implications.
Spokesman Nicola Duckworth said: "No-one should be deported to face a risk of torture, whatever they might be alleged or suspected to have done.
"States simply cannot pick and choose which people have human rights.
"If these individuals in question are reasonably suspected of having committed a criminal offence relating to terrorism, it is always open to the UK authorities to charge them and give them a fair trial."