The crisis also has implications for Washington, which wants a European Union that can promote common interests in places like Afghanistan and the Middle East with financial and military help.

“All of that is in doubt if the cornerstone of the E.U.  its internal market, economic union and solidarity  is in question,” said Ronald D. Asmus, a former State Department official who runs the Brussels office of the German Marshall Fund.

The problems are basically twofold: within the inner core of nations that use the euro as their common currency, which together have an economy roughly the size of the United States’; and within the larger European Union.

The 16 nations that use the euro  introduced in 1999, and one of the proudest European accomplishments  must submit to the monetary leadership of the European Central Bank. That keeps some members hardest hit by the economic downturn, like Ireland, Spain, Italy and Greece, from unilaterally taking radical steps to stimulate their economies.

Germany once vowed never to bail out weaker members in return for giving up its strong national currency, the deutsche mark. But German leaders are now faced with the unpalatable prospect of having to put German money at risk to bail out less responsible partners that do not adhere to European fiscal rules.

Within the larger European Union, fissures are growing between older members and newer ones, especially those that lived under the yoke of Soviet socialism. Some countries of Central Europe, like the Czech Republic and Poland, are doing relatively well. Others, including Hungary, Romania and the Baltic states, are in a state of near-meltdown.

Image The leaders of European Union countries who gathered Sunday in Brussels included, from left, Sergei Stanishev of Bulgaria, Nicolas Sarkozy of France, Andrus Ansip of Estonia, Demetris Christofias of Cyprus, Mirek Topolanek of the Czech Republic and Lawrence Gonzi of Malta. Credit... Olivier Hoslet/European Pressphoto Agency

But only two newer members  tiny Slovenia and Slovakia  are protected by being among the countries that use the euro, and there was little support on Sunday for changing the rules to allow more to join quickly.