The two-day summit, due to start on Friday, will be fighting on two fronts.
The first is short term. Leaders must defuse the immediate threat as bond markets pick off weaker, debt-burdened economies and banks, fearing a cataclysm, stop lending to each other - in essence, halting the flow of blood to the financial system.
Chancellor Angela Merkel of Germany, the EU paymaster, is under huge pressure to let the European Central Bank become a lender of last resort, issuing euro-bonds to underwrite sovereign debts.
Merkel is fiercely opposed to this, but may signal some stop-gap measures, such as short-term bonds to support the eurozone's bailout fund.
The other big challenge is long-term. The summit must restore the credibility of the euro's governance after a crisis that has ripped Europe open along old but invisible faultlines.
It exposed the lies of successive Greek governments, who deceived Brussels about their astronomical borrowing. Spain's central government was found to have lost control over its regions, which gorged on cheap debt. And Italy's corrupt political system and Belgian's linguistic civil war shed light on chaotic coalitions unable or unwilling to confront an economic storm.
All these inadequacies dragged down the euro. Hastily assembled bailouts became bigger and bigger, confidence in the currency flagged and the bond markets hiked the cost of borrowing to runts in the litter to unsustainable rates.
Fixing these deep-rooted problems has caused thinking to polarise around two possible solutions. One is an orderly return to national currencies, or perhaps a "hard" euro that would gather only a handful of well-run northern economies, led by Germany. Right now, these options are viewed with dread. The softer southern economies would be crippled by past debts denominated in euros. And the harder northern economies, whose currency would rise in value, would see their exports badly hit.
The other solution, driven by Germany and France, the traditional locomotive for European integration, is to overhaul the eurozone rulebook so the borrowing madness never happens again. "Rules must be respected. Respect for them must be supervised. Their violation must have consequences," Merkel told Parliament last week.
What this will mean in practice - and how to achieve it - will be fleshed out in the next few days. A diplomatic waltz on adrenalin begins with a Franco-German summit tomorrow, after which the two leaders will be racing around Europe to secure support from other governments.
"EU leaders have until their summit meeting next weekend to come up with something investors can believe in. Until then the euro is on parole," London-based financial analysts Moneycorp said.
Merkel called for a "fiscal union" where sanctions would be cast in stone, made legally binding in "constitutional" form and enforceable by the European Court of Justice, the EU's top judicial body. She also wants a specially appointed European commissioner to intervene directly in national budgets when deemed necessary.
Sarkozy, though, is wary of the erosion of sovereignty that comes from handing additional powers to the European executive and top court, according to sources in Paris. Other leaders may well agree. Modifying or creating new treaties is a political minefield in Europe, as this requires ratification by all members.
Diplomats in Brussels say the summit will be urged to endorse a change under Protocol 14 of the Lisbon treaty, which affects only eurozone countries. If this fails, Germany and France could go for an inter-governmental agreement on the lines of the Schengen border-free pact.
It would involve the 17 eurozone members, or just those willing to sign up a scenario that opens the way to a European economy to an inner core and an outer circle. Europe's crunch week may well end in tiers - or tears.
