Greek Prime Minister Alexis Tsipras (front) attends the St. Petersburg International Economic Forum in St. Petersburg, Russia, on Thursday. (Maxim Shemetov/Reuters)

GREECE HAS appeared to be on the brink of collapse so many times over the past half-decade that observers on this side of the Atlantic can perhaps be forgiven for tuning out the whole endless “crisis.” Surely cooler heads will prevail; a bailout will be cobbled together by Germany and the International Monetary Fund; Greece will limp along but not exit the European common currency or precipitate a wider catastrophe for the world economy. It has worked out that way so far, hasn’t it?

Maybe not this time. With its current bailout package expiring June 30 and a $1.7 billion payment due to the IMF that same day, Greece desperately needs a fresh infusion of aid. The IMF recommended aid, including looser budgetary conditions and long-term debt relief, yet the minimum reforms it seeks in return exceed the maximum that the leftist government in Athens is willing to promise. There is still hope of a breakthrough when the Greek prime minister, Alexis Tsipras, meets German Chancellor Angela Merkel at the European Union summit on June 25. But the Greek central bank thinks the prospects of an impasse are so high that it has publicly warned of “an uncontrollable crisis” ahead.

That forecast is all too plausible for Greece itself, which would probably have to impose capital controls and pay government creditors with IOUs, while living standards continued to fall and the nation’s best and brightest emigrated in search of work. The wider repercussions are less predictable. Thanks to aggressive action by the European Central Bank, coupled with reforms already undertaken by their governments, other debtor nations in Europe such as Spain and Italy might be able to resist any “contagion” from a Greek bankruptcy. Certainly, Ms. Merkel will be more inclined to help them because they are still governed by relatively moderate mainstream politicians, in whose success she has a stake — just as she has an interest, albeit an unstated one, in the populist Mr. Tsipras’s failure.

In fact, the political aspects of a Greek default could prove even trickier, and for the rest of the world, more consequential, than the economic ones. If you think Russia’s Vladi­mir Putin wouldn’t try to exploit a split in Europe, starting with Greece, think again. Mr. Putin has long sought to undermine the unity of NATO and the European Union, including through political flirtation with right- and left-wing populists across the continent. Mr. Tsipras, who has pointedly criticized Western sanctions on Russia for its aggression in Ukraine, already acts too much as Mr. Putin’s apologist within the Atlantic alliance. A rupture between Greece and the rest of Europe might exacerbate that tendency.

The goal for Europe’s leaders must be to exhaust every option for avoiding the “uncontrollable” situation of which they have been warned. Failing that, they must do even more to shore up the rest of the continent’s troubled economies, to demonstrate to their people the tangible benefits of the European project. That project failed badly in Greece; it can still be redeemed if the continent’s leaders draw the right lessons from this bitter experience.