Home Secretary Theresa May has used her first major intervention in the EU referendum debate to argue that Britain should stay in the 28-nation bloc but leave the European Convention on Human Rights (ECHR).
Mrs May's comment raised the prospect of continued wrangling over Europe even after the June 23 vote. And it came as prominent Brexit backer Owen Paterson warned that the Government would not be able to put the "genie" of EU withdrawal "back in the bottle" if there is a narrow vote to Remain.
Meanwhile, US president Barack Obama followed up his warnings on the economic risks of Brexit with an appeal for a "united" Europe to help drive global security, democracy and prosperity.
Speaking in the German city of Hanover ahead of a summit with David Cameron and other EU leaders, the American leader acknowledged that European unity involved "frustrating compromise", but hailed the multinational union as "one of the greatest political and economic achievements of modern times".
Mr Obama said: "I've come here today to the heart of Europe to say that the United States and the entire world needs a strong and prosperous and democratic and united Europe.
"A strong, united Europe is a necessity for the world because an integrated Europe remains vital to our international order."
Mr Obama's intervention came as Brexit's biggest hitters sought to seize back the referendum initiative by putting immigration at the top of the agenda.
Justice Secretary Michael Gove warned the UK faces a migration "free-for-all" unless it breaks away from Brussels, while London mayor Boris Johnson said that Mr Cameron had achieved "two-thirds of diddly squat" in negotiations for a special deal for Britain on immigration and other key demands.
But Mrs May insisted that "nobody should think" Brexit is the "single bullet that is suddenly going to solve all our immigration problems".
While free movement rules "mean it is harder to control the volume of European immigration", they "do not mean we cannot control the border," she said, in a speech in London.
Co-operation with other EU states on criminal records, biometric data and the European Arrest Warrant helped improve Britain's safety, but the ECHR - which is an agreement of the Council of Europe and not the European Union - "makes us less secure by preventing the deportation of dangerous foreign nationals", she said.
Mrs May added: "So regardless of the EU referendum, my view is this: if we want to reform human rights laws in this country, it isn't the EU we should leave, but the ECHR and the jurisdiction of its court."
In his Daily Telegraph column, Mr Johnson warned the Remain side not to "crow too soon" that the Leave side had been "bombed into submission".
"The Prime Minister asked the EU for reform and got two-thirds of diddly squat. That deal shows how contemptuously we will be treated if we remain," Mr Johnson said.
Former work and pensions secretary Iain Duncan Smith said immigration was "out of control" and "poorer people" had seen their livelihoods damaged by EU citizens coming to the UK.
And Mr Gove warned that the possible future accession of countries like Turkey and Albania would mean public services like the NHS facing "an unquantifiable strain as millions more become EU citizens and have the right to move to the UK".
Writing in The Times, the Justice Secretary warned: "We cannot guarantee the same access people currently enjoy to healthcare and housing if these trends continue. There is a direct and serious threat to our public services, standard of living and ability to maintain social solidarity if we accept continued EU membership."
But Downing Street said any discussions on new EU members were "years away" and that the UK had a veto over future enlargement. Mr Cameron's renegotiation deal ensured that different transitional rules would be applied to nationals of any future member states, said the PM's official spokeswoman.
And Mrs May signalled a break with the UK's long-standing support for Turkish membership, asking whether now was the right time "to contemplate a land border between the EU and countries like Iran, Iraq and Syria".
Former cabinet minister and EU commissioner Lord Mandelson said the Vote Leave campaign had "hoisted the white flag on arguments around the economy" and were now running a "Ukip-lite strategy centred on immigration".
Mr Paterson insisted that he still believed Leave would win and warned the UK would be relegated to the status of a "colony of Europe" if it remained within the EU.
Even a narrow victory for Remain would mean millions of voters supporting Leave, showing that anti-EU sentiment was a "very respectable" mainstream opinion rather than a view held by "nutters" on the fringes, he said.
"Millions of people from right across the country from every class and occupation will have stated that they want to leave the European Union. You won't put that genie back in the bottle," said the former environment secretary.