Politics, a long-deceased British statesman once said, is the art of the possible. For Gordon Brown, who takes over as the U.K.'s prime minister on June 27, politics is going to be the art of the impossible.

Imminently, Mr. Brown will be faced with an intractable dilemma. Call it a treaty, call it a constitution, call it a series of amendments to existing treaties, but something is going to emerge from this week's European Union summit. And how Mr. Brown reacts to that document will decide his political future. But whatever he does will be the wrong decision.

Those behind the revival of Valéry Giscard d'Estaing's 2004 EU constitution do not, of course, want to allow pesky voters to reject their renewed instructions. The French and the Dutch electorate behaved intolerably once before in rebuffing the wisdom of their political masters and cannot be allowed to interfere again. French President Nicolas Sarkozy says he will not put the treaty to a referendum. Fellow European leaders will pressure Mr. Brown to nod through whatever Tony Blair agrees to this week in Brussels.

There is no constitutional requirement in Britain for a referendum. Mr. Blair's promise of one on the earlier treaty was unprecedented, and Mr. Brown could argue that this year's version is not as extensive and thus doesn't require a popular vote.

He could do that. If he did, however, he would confirm at the very outset of his premiership all the electorate's worst images of him as a Stalinist control freak, as a man contemptuous of opposing views, and as a politician unworthy of the country's highest office.

Worse, while there was no constitutional requirement for Mr. Blair to put the previous treaty to a referendum, his decision set just that precedent. And going by precedent is the legal nostrum at the heart of Britain's unwritten constitution.

Denying a referendum would also unleash the mother and father of all attacks from the media and the public. Not a single newspaper would be likely to support such a decision, and the most powerful papers -- the Daily Mail and the Sun -- would rip Mr. Brown to shreds. The Sun's support for Labour has been the bedrock of the party's media strategy. Mr. Brown has invested countless hours in courting the editor of the anti-Labour Mail; successfully so, given the paper's recent praise for his sagacity and probity. Ruining that relationship would be monumentally self-defeating.

Not calling for a referendum would also, quite possibly, be like conceding the next election to the Tories. For the first time since he was elected leader of the Conservative Party in 2005, David Cameron is facing real hostility from his members over his renunciation of academically selective "grammar" schools, a favorite cause of traditional Tory activists. There is, however, one topic on which his party remains overwhelmingly united: its opposition to any further weakening of British sovereignty and any moves toward a European state. Were Mr. Brown to deny the British public a vote, all of Mr. Cameron's Christmases would have come at once. Not only would the Conservative leader have an issue with which to enthuse and unite a party still wary of his leadership, he would also have an issue on which he could (rightly) claim to speak for the nation.

That is the nub of Mr. Brown's impossible choice. And if he did do right by the voters and put the treaty to a vote, he would also be signing its death warrant. Voters would certainly reject the treaty -- any treaty. As the EU's own Eurobarometer research shows, only 34% of Brits agree that EU membership is "a good thing" -- the lowest figure of any EU member. The idea that Britain would support a further integration is a nonstarter.

So from the moment Mr. Brown announced a referendum, the treaty would be finished. And what a start that would be for his relationship with fellow EU leaders. Never mind that he would be doing the EU a favor -- rescuing it from taking yet another step away from voters and legitimacy. The U.K. leader would be treated like poison by Mr. Sarkozy and German Chancellor Angela Merkel. They would know that Mr. Brown had not been hijacked by a surprise result, as befell former French President Jacques Chirac. Rather, the new U.K. prime minister would have deliberately chosen a course of action leading only to defeat.

Anyone who has witnessed Mr. Brown's forays into EU meetings will know that he has almost no interest in the EU. His view is that Europe's leaders can do what they want; he has nothing to learn from them and will carry on doing what he knows is right for Britain. Caring little for grandiose visions of a European dream, he presumably will put domestic U.K. politics ahead of any wish not to rock the EU boat and thus call a referendum.

Except for one thing. A referendum defeat -- the certain outcome -- is, after all, a defeat. Even if his support for the treaty is lukewarm, Mr. Brown will not want to begin his time in office with a humiliating loss at the polls. He could not, after all, call a referendum seeking support for a treaty and then distance himself from it. If his support were anything other than wholehearted he would look vacillating and weak. And the opposition would run rings around him, accusing him of foisting on the country a treaty which he did not back.

That is Mr. Brown's impossible dilemma. There are only two choices open to him. Either will be disastrous.