David Cameron faces the most severe challenge of his career and only Labour can solve it for him. The campaign to keep Britain in the EU can be won only if enough Labour supporters cast a ballot to stay in.

This under-appreciated twist in the referendum plot puts the prime minister in the deliciously ironic position of looking to Labour to save him from the wishes of much of his own party. It also places a momentous burden of responsibility on Labour. In normal times, what the opposition says usually makes a marginal contribution to the destiny of the nation. In the extraordinary circumstances of the referendum, the future of the United Kingdom and its place in the world is in the hands of Labour and its supporters to a degree few have fully appreciated.

To see why, start with the maths. We know that a big chunk of the Tory vote, along with almost all of the Ukip vote, will be casting a ballot for Brexit. This means that what happens to the Labour vote will be pivotal. If Britain is to remain inside the EU, the largest element of the coalition to remain will have to be drawn from people who are always or usually Labour supporters. One strategist for the In campaign says: “The Labour vote is critically important.” I’m tempted to go further and say the Labour vote will decide the outcome.

On the face of it, the referendum campaign ought to be an opportunity for Labour to present a united face to the world. That would make a nice change. While Tories talk in many tongues on Europe, Labour should be able to speak with one voice. While there are differences of emphasis around the edges, on the big question – remain or leave? – the party is united. The vast majority of Labour MPs, more than 200 of them, support continuing EU membership. Fewer than 20 Labour MPs have declared for Out.

While some of David Cameron’s cabinet will campaign against their prime minister, there is no one in Labour’s senior team advocating departure. The great majority of Labour members are pro-membership, including those who describe themselves as Corbynistas – or so polling of their attitudes suggests. This unity does, though, come with a snag. Labour activists do not feel anything like as animated about Europe as their counterparts on the right. While the question is of epic significance for the UK, it has not been a cause of burning passion within Labour’s ranks since the late 1980s when the party began to move into a more settled pro-European position under Neil Kinnock and his successors. Many Tories live, breathe and define themselves in terms of Europe; hardly anyone in Labour feels the same.

So one pressing concern for In campaigners is that this is not seen simply as “Cameron’s referendum”. That would make it much harder to motivate Labour activists and voters. Some might well feel inclined to vote Out as a way of giving a kick in the ballots to a Tory PM. To stop that happening, the messages their leaders send to Labour voters are going to matter a lot. They have to be convinced that Britain should stay and further persuaded that the question is of such importance that they need to turn up on the day of decision and vote to remain even when the proposition is being put by the enemy party leader.

The Tory leader may have started this process, but it will be Labour people who decide how it ends

This makes the posture of the chief of the Labour tribe important. How he behaves is bound to influence the amount of zeal, or the lack of it, that party activists display in campaigning, as well as shaping the responses of Labour voters. There remains a question mark about the attitude of Jeremy Corbyn, a question mark that is disturbing the sleep of some of his senior colleagues. Early in his leadership, the shadow cabinet corralled him into declaring that Labour would campaign to remain in the EU whatever sort of deal Mr Cameron came up with.

Colleagues don’t detect signs of Mr Corbyn resiling from that commitment, but some are becoming jittery that he is displaying scant enthusiasm for the cause. This is not altogether surprising. It would be consistent with the Labour leader’s history and beliefs to regard the EU as a “capitalist club” and therefore not something worth fighting for. Alan Johnson, the senior figure in the Labour Yes campaign, last week went along to the shadow cabinet to brief them about the prospects for the referendum. Several around the table expressed their full-hearted support for his efforts, but not Mr Corbyn. I’m told the Labour leader said very little at all. It also concerns some of his colleagues that the Labour leader has yet to agree to appear at any campaign events. He doesn’t have to actively oppose an In vote to do harm to the In cause. He only has to radiate a shrugging attitude to the great question and it could be read as a message to Labour supporters that it is OK for them to vote for Brexit or not to bother voting at all.

If the Labour leader proves reluctant to sound the trumpet, then the tunes played by other senior figures in the Labour orchestra will have added importance. Mr Johnson is universally seen as a persuasive figure with Labour voters. That was why he was selected to front the campaign in the first place. Some of the better-known figures in the shadow cabinet may also be influential with Labour voters. Of the older generation, ardent pro-Europeans like Tony Blair may not be that helpful to the cause because their views are so unsurprising that they are discounted. Figures who can connect with northern, working-class voters and talk in practical terms about the advantages of EU membership and the risks of being outside – the name of David Blunkett is often mentioned– will be important in motivating the Labour vote.

The most immediate decision that Labour will have to make is how to react when David Cameron presents his final deal. That will be on Friday if – that’s still quite a big if – all goes to plan at the European Council meeting. About how to respond, there is an unresolved argument within the shadow cabinet. Ever since the Tory leader embarked down this road, it has been a default Labour attack line to say that the entire thing has been about managing the Tory party rather than promoting the interests of the country. This has had the merit of being true, but it is a narrative that will be both irrelevant and trivial once the campaign is under way and Britain’s future is at stake. Some Labour figures remain attached to reflexive oppositionalism and want to rubbish the Cameron renegotiation. Others will broadly smile on his deal – or at least pretend that they think he has achieved something worthwhile in the belief that sounding positive will assist the over-arching cause of keeping Britain within the EU.

There is also a building clash about how Labour ought to talk about specifics of the Cameron deal. The sharpest tension is over the “emergency brake” on the benefit rights of migrant workers from other EU states. The strong instinct of Jeremy Corbyn is to oppose this element of the Cameron package on the grounds that it is discriminatory and all workers should be treated equally. As we report today, the Labour leader is planning an intervention to that effect, the timing of which has not yet been precisely fixed. This puts him at odds with a lot of his shadow cabinet. “That doesn’t sit well with Andy Burnham,” says one master of understatement in Labour’s senior team. Several of them are aghast to learn that this is the element of the deal that their leader wants to make the focus of his fury. Their encounters with voters tells them that anxiety about immigration is the single most important ingredient fuelling hostility to the EU.

The hopes of Labour’s pro-Europeans are pinned on the idea that other considerations among their voters will be ultimately trumped by fears about what will happen to the economy if Britain exits the EU. Labour frontbenchers are being encouraged to memorise the acronym JISI: Jobs, Investment, Security and Influence.

Work has begun on the ground campaign by those Labour people who appreciate the vertiginous scale of the stakes. The sheets used by Labour canvassers for the May elections recently had an extra question added to them, which asks about voting intentions in the referendum. It is also gradually dawning that the referendum campaign will be quite unlike a general election and fought to entirely different rules. In a general election, the government is chosen by swing voters in marginal seats. In the referendum, every vote has equal weight. Voters in safe seats are every bit as important as voters in marginals. It will be a particular challenge persuading Labour folk to cast a ballot in the referendum when they live in seats so safe that many have got out of the habit of voting.

For the only time in his life, David Cameron will be desperate for a Labour campaign to be a stonking success. His place in history and Britain’s place in Europe depend on it. The Tory leader may have started this, but it will be Labour people who decide how it ends.