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* Britons more sceptical towards European Union
* Concerns that UK edging towards the exit
* "We're either a union or we're not", says Irish minister
By Luke Baker
DUBLIN, Jan 8 (Reuters) - Ireland, which holds the presidency of the European Union for the next six months, urged Britain not be too adversarial in pushing for change in its relationship with the EU on Tuesday, saying it should remember its close history in Europe.
Prime Minister David Cameron, responding to growing Euroscepticism among voters at home, has said he wants to overhaul Britain's 40-year-old links with Brussels and is expected to flesh out his thinking in a speech later this month.
His increasingly vocal insistence on forging a new contract with the EU has given rise to concerns that Britain, which joined the now 27-country bloc in 1973, is edging towards an exit - a moved dubbed "Brexit" - even though Cameron himself has said he thinks Britain should remain part of the EU.
Britain's position has to a large extent replaced the threat of Greece leaving the euro zone - a possibility once dubbed "Grexit" - as the main focus of political uncertainty in Europe.
"Speculation about a Greek exit has thankfully stopped, but now there is talk about the UK," Lucinda Creighton, Ireland's minister for European affairs, told Brussels-based journalists visiting Dublin as Ireland assumes the EU's presidency, a largely ceremonial role that it will hold until July.
"The relationship between the UK and the EU is a big challenge and it is one we and the British government will have to deal with sooner or later," she said.
"I hope David Cameron will reflect on and recognise the important role that the UK plays in the European Union and the important relationship that the EU has with the UK."
"A LA CARTE MENU"
Ireland's deputy prime minister and foreign minister, Eamon Gilmore, reiterated Creighton's line, saying Britain and Ireland shared close historical, trade and investment ties and Dublin would do what it could to keep London engaged in Europe.
"With the kind of world we live in now ... the direction that we have to be going is towards working together rather than separating," he said, describing Ireland's current relations with Britain as better than at any time in their history.
But referring to Cameron's wish to opt out of large parts of EU legislation and then decide what to sign back up to, he said: "The European Union is not an a la carte menu."
"We're either a union or we're not. This is not going to work if we have 27 or 28 categories of membership."
Britain's relationship with Europe has grown increasingly rocky over the past three years, with a low point touched in late 2011, when 25 of the EU's 27 member states agreed on a new fiscal treaty, leaving Britain and the Czech Republic out.
Tensions are particularly high over financial regulation, with Britain concerned that rules drafted in Brussels could limit the influence of its powerful banking and finance sector. There are also deep differences over employment law and legislation concerning justice and policing issues.
Ireland, which joined the European Economic Community, the forerunner to the EU, on the same day as Britain in 1973, said it would do what it could to liaise with its neighbour across the Irish Sea to ensure it stayed close to Europe.
"We're a neighbour and we're a friend of the UK. We have a deep interest in ensuring that the UK stays engaged," said Creighton.
But asked about British voters' increasingly uneasy feelings about the EU and the possibility of a referendum on EU ties being held in the coming years, she added: "We would respect whatever decision the British people make."