The talks on Monday addressed the firewall issue, though only briefly. A new, permanent fund of 500 billion euros, or $660 billion, called the European Stability Mechanism, is due to come into existence in July, and one way of bolstering its power would be to run it alongside the current, temporary, rescue fund, the European Financial Stability Facility.

The bailout, with a stronger firewall, could provide the euro zone with some much-needed momentum. The injection of liquidity into the banking sector by the European Central Bank late last year — with the tacit support of Germany — had started to convince critics that there was a determination to save the currency.

Yet only last week the Greek bailout appeared to hang in the balance when rumors circulated that Germany’s finance minister, Wolfgang Schäuble, was willing to contemplate a Greek default. As tempers flared, the Greek finance minister, Evangelos Venizelos, suggested that some people were trying to drive his country out of the euro zone, and the Greek president, Karolos Papoulias, accused Mr. Schäuble of insulting the country.

It remains unclear whether a default was contemplated seriously or merely floated as a means of pressuring Athens.

Nevertheless, the episode underlines the extent to which Greece remains a weak spot for the 17 European Union countries that use the euro. This would be the second major bailout for Greece in two years. In May 2010, European governments and the International Monetary Fund put together the first three-year bailout package of 110 billion euros, then worth about $146 billion, not all of which has been used.

Doubts persist about Greece’s ability to carry out the tough austerity measures pushed through Parliament or to manage the weakened economy.

Those long-term fears deepened with the leak of an official report from the European Commission, European Central Bank and the International Monetary Fund that said that if changes were not made, Greek debt could remain at 160 percent of G.D.P. in 2020. It also suggested that more help would be needed after the period covered by the bailout being negotiated. That could amount to $66 billion more from 2015 to 2020.