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DAVOS, Switzerland — British Prime Minister David Cameron wants nothing to do with a United States of Europe, an idea that’s gaining currency as the countries that use the euro struggle to fix their debt crisis.

But what if it’s a choice between a single country called Europe or a splintered continent? Cameron is determined to avoid that scary scenario.

A day after he shook up Europe’s political landscape by offering British citizens the prospect of a vote on whether to stay in the 27-country European Union, Cameron insisted Thursday he wants Britain to remain a part of the bloc but that more unification would not be the answer.

Many in the EU, particularly among the 17 countries that use the euro, are on a drive for closer political unification, and that’s raised particular concerns recently in Britain, which has often viewed the bloc through a business prism.

After decades of bit-by-bit integration, the links are now so tight that many European leaders refuse to publicly acknowledge that a British exit is a possibility.

Mark Gray, a spokesman for the EU, said the bloc affects almost all aspects of the lives of Britons, from the quality of the water they swim in at beaches or in pools, to the quality of the orange juice they have for breakfast and the conditions in the offices where they work.

But many Britons — like citizens elsewhere in the EU — see the union as a faceless beast imposing rules and spending on needless things and threatening sovereignty.