Mr Cameron reiterated a pledge contained in last year's Conservative election manifesto to repeal the Act if it could not be successfully reformed.
Human rights legislation has come under sustained attack this week because of two unrelated cases, one involving a murder by a man who had been released after spending 16 years in prison for rape, and the other involving a group of Afghan hijackers who have been allowed to stay in Britain.
Mr Cameron's intervention has won him support from Britain's biggest selling newspaper, The Sun - a prize coveted by every opposition leader - but could come at the price of undermining his own attempts to change the image of the Conservative Party.
Mr Cameron told yesterday's Sun that it is "close to impossible" to deport foreign nationals who pose a threat to the UK. He added: "It is wrong to undermine public safety by allowing the human rights of dangerous criminals to fly in the face of common sense. The Government's attitude has been complacent. It refuses to recognise that the problem is compounded by the interpretation of the European Convention and the passage of the Government's own Human Rights Act."
A spokesman added later that Mr Cameron's policy could also involve Britain "temporarily" withdrawing from the European Convention, to renegotiate some of its clauses.
The Government is currently negotiating with Middle Eastern and north African countries such as Jordan, Lebanon and Algeria to obtain guarantees that any suspects returned there will not be tortured or killed, so that their nationals can be deported without breaching international law.
"If the Government succeeds in its attempts to achieve memorandums of understanding, then so much the better. But if not, we will reform, replace or scrap the Human Rights Act," Mr Cameron said.
But Shami Chakrabarti, director of the civil rights pressure group Liberty, warned: "The Conservatives will have to decide whether they are going back to being the nasty party, or whether they are the Cameron party. Are they still the civil rights party, as they have been recently when they opposed control orders and ID cards?"
The former Europe minister, Denis MacShane, said: "Cameron seems to have said this for the sake of a headline without thinking it through. You can tear up the British legislation, but that only means that cases will go to the European Court of Human Rights, so it doesn't actually do anything. But if Cameron is saying that Britain is going to withdraw from an international treaty, we will be out of line with every other civilised country."
Nine Afghans hijacked a Boeing 727 in the capital, Kabul, in February 2000 and forced the crew to fly to Stansted, in Essex. They were escaping from the Taliban regime. A judge ruled this week that the nine, who cannot be named for legal reasons, should be allowed to stay in Britain. The Home Secretary, John Reid, has described the ruling as "inexplicable" and is seeking to overturn it.
The Prime Minister's official spokesman indicated yesterday that it was the way the Human Rights Act is applied by the courts, rather than the legislation itself, which was causing public concern. "There is a need to reassure the public that common sense logic should be right through the legal system, so there is the need, if necessary, to rebalance the system so that happens," he said.
Separately, an inquiry by the Chief Inspector of Probation, Andrew Bridges, into the murder of Naomi Bryant by Anthony Rice, who had been released from a life sentence for rape nine months earlier, had warned that staff were sidetracked into considering Rice's human rights rather than protecting the public.
Ten facts about the law
* The 1998 Human Rights Act incorporated the European Convention on Human Rights into British law. It meant that British courts heard cases which previously would have gone to the European Court in Strasbourg.
* Neither the convention nor the European Court is linked to the EU or Brussels. The ECHR is a separate international agreement signed by 46 countries, including Russia and Turkey.
* The EU has its own European Union Charter of Fundamental Human Rights, but this is a political statement that is not directly enforceable in law.
* The Human Rights Act does not overrule other laws passed by Parliament. A judge can declare that a piece of legislation is incompatible with the HRA, but that does not invalidate it.
* The HRA cannot therefore prevent Tony Blair from passing laws to deport all foreign prisoners on completion of their sentences, or to ban all hijackers from being admitted to the UK - although if he did, the laws might be challengeable under the ECHR.
* The Act does not only protect offenders. It protects the privacy and the right to compensation of victims and of the bereaved.
* It does not only apply to the police and law courts. For example, if you go into an NHS hospital, your rights there are protected by the RHA.
* Under certain circumstances, it also bans discrimination at work on the grounds of sex, race, religion or other factors such as disability.
* If your employer invades your privacy, for example by intercepting your telephone calls, or prevents you joining a union, or if you are forced to join a union, you can invoke the RHA at an employment tribunal.
* Freedom of speech was first enshrined in law under the Human Rights Act. Previously, it was a 'negative' right - you could exercise it unless there was a law stopping you.
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