When British Prime Minister David Cameron decided to call a referendum on the U.K.’s membership in the European Union, he did so in the belief that it would settle the question once and for all. Deep divisions over the issue in Britain and within Mr. Cameron’s own party were undermining not only the U.K.’s relationship with the EU, but the stability of the EU itself. Confident he would win, Mr. Cameron believed the referendum would heal the division in the Conservative Party and allow the U.K. once again to play a constructive role in European affairs.

With just 53 days to go, this conventional wisdom is starting to look naive. That is partly because Mr. Cameron may yet be heading for defeat, despite scoring several points in the past couple of weeks. Though a series of recent heavyweight independent analyses have predicted a British exit, or Brexit, would have a negative long-term impact on the U.K. economy, and President Barack Obama warned a post-Brexit U.K. shouldn’t expect any quick U.S.-U.K. trade deal, the polls suggest the vote remains on a knife-edge. The latest poll of polls, which averages the six most recent polls, shows the Remain campaign ahead by 51% to 49%, according to whatukthinks.org. That is too tight for comfort for the Remain camp since Brexit supporters, who tend to be older, are thought much more likely to vote than younger Britons.

But even if Mr. Cameron wins, it seems increasingly unlikely the referendum will settle anything. That partly reflects the tone of the debate, which has so far been presented as a choice between being half in or fully out of the EU, says Anand Menon, professor of European politics and foreign affairs at King’s College London. The Remain campaign has emphasized Britain’s special status in the EU, including its opt-outs from the eurozone, the Schengen passport-free zone and Mr. Cameron’s recently won exemption from the EU’s commitment to an “ever-closer union.” There has been little attempt to appeal to any sense of common European interests or identity. That suggests that the best that the Remain campaign will be able to claim is grudging British acceptance of EU membership, rather than a ringing endorsement.

Meanwhile, the referendum looks likely to leave a lasting scar on the British political landscape. The Conservative Party is more split than Mr. Cameron ever expected: Roughly half of his parliamentary colleagues are backing Brexit and so are an estimated 70% of the party’s grass-roots members.

That makes it likely that the next leader of the Conservative party will be a Brexit supporter. Mr. Cameron is due to stand down before the next election in 2020, but there is speculation he will face a leadership challenge sooner, regardless of the outcome of the referendum.

At the same time, the U.K. Independence Party, Britain’s only avowedly euroskeptic party, is determined to use the referendum to make further inroads into the Labour Party’s core working-class heartlands. UKIP is expected to make further gains in local elections this week at the expense of Labour, which is itself badly split over allegations of anti-Semitism and internal opposition to its new left-wing leadership.

What is clear is that to have any chance of settling the question of the U.K.’s European destiny, Mr. Cameron must not only win, but win by a comfortable margin—which some Conservative allies define as a 60/40% vote for Remain. Brexit campaigners are already preparing the ground for a “stab-in-the-back” narrative should they lose, complaining loudly that Mr. Cameron is attempting to rig the referendum by using the government machine to lobby voters.

Only a decisive victory would give Mr. Cameron a strong mandate to reassert control of his party and rebuild trust with the rest of the EU, playing a constructive role in tackling common challenges facing the continent. A narrow victory would raise the prospect of a “neverendum,” as in Scotland, where a 55/45% vote against independence in 2014 hasn’t been sufficient to remove talk of a second referendum.

A decisive victory isn’t yet out of the question. Remain campaigners are confident they have won the economic argument, which polls suggest is the most important issue for most voters. But to be sure of success, Mr. Cameron needs to persuade people to vote, to turn grudging support for the EU into enthusiastic backing. That will likely require more than simply highlighting the risks of exit—what Brexit campaigners have dubbed Project Fear. He will need to set out a positive case for continued EU membership, explaining how a post-referendum U.K. might work with the rest of the EU to address voter concerns including terrorism, illegal migration, instability on Europe’s borders, stagnant living standards and the social strains caused by globalization.

For a prime minister who until a few months ago wasted few opportunities to talk down the EU, this change of tone is certainly a challenge. Far more than his own fortunes depend upon him rising to it.