Greece lost its financial lifelines Tuesday, as the country missed a crucial payment to the International Monetary Fund amid growing questions about whether it would be able to remain in the euro zone.

Greek leaders had made a last-ditch attempt to come up with the necessary cash, asking European countries for a new bailout hours before its last ones were set to expire, but E.U. finance ministers rejected the request as unrealistic. The missed payment, confirmed by the IMF, was a landmark moment in Europe’s five-year battle to preserve its common currency.

The E.U. finance chiefs were set to reconvene Wednesday as Greece’s cash dwindles and its banks remain closed. The ministers’ decision to hold firm was a sign that they believed they had successfully put in place the defenses­ against instability in Europe if a country left the euro zone. But as Greece became the first developed nation to miss a payment to the IMF, E.U. leaders were confronting the prospect of a European country plunging into intense financial misery as it was forced to abandon the currency.

“Greece is fully committed to servicing its external debt in a manner that secures the viability of the Greek economy, growth and social cohesion,” Greek Prime Minister Alexis Tsipras wrote in a letter ahead of the missed payment, requesting the new bailout. It was addressed to Dutch Finance Minister Jeroen Dijsselbloem, who leads the group of 19 finance ministers of euro-zone countries.

Greeks are strongly in favor of remaining with the euro. But they have rebelled against the crushing austerity measures that Europe has demanded in exchange for bailing out the indebted government. Tsipras stunned E.U. leaders last weekend by calling for a July 5 referendum in which he intends to put the European demands to the voters’ test. He has campaigned for a “no” vote, which he says will strengthen his bargaining power with his E.U. counterparts.

Those same leaders have said that such a vote would amount to a decision to leave the 19-nation euro-currency union. E.U. finance chiefs took a hard line on Tuesday’s new bailout request, saying that E.U. nations had the right to request assistance but that any request would be addressed through ordinary, slower channels.

“Politically the situation hasn’t changed. There is no new ground,” Dijsselbloem said after the E.U. finance ministers consulted Tuesday. “The situation in Greece, in the Greek economy, in the Greek banks, has deteriorated even more, unfortunately.”

The $1.67 billion missed payment to the IMF was unlikely by itself to spur immediate problems for the global economy, since it affected only a government-backed institution, not private investors. But if Greece is ultimately forced off the euro, other troubled euro-zone economies such as Portugal could be seen as more vulnerable. The exit could also weaken the goal of ever-closer European integration.

In Washington, President Obama told reporters that the United States is watching the Greek debt crisis with “substantial concern” but that the fiscal woes should not bring “a major shock” to global financial networks. He spoke to German Chancellor Angela Merkel over the weekend to urge that a way be found for Greece to remain in the euro zone.

In Greece, meanwhile, the economy was near frozen. Fearing a panicked run to withdraw euros, the government has closed banks and set a withdrawal limit of 60 euros, about $67, per day at ATMs. Retirees, who usually withdraw their pensions in person at banks, have been particularly hard-struck by the restricted access to financial institutions. The government has said it would sharply limit their withdrawals but give them access to cash later in the week.

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Tsipras’s request was for a two-year emergency bailout from the European Stability Mechanism, a fund that was created after the first series of Greek emergency bailouts started in 2010. He asked for $32 billion to help pay debts due through 2017.

Tsipras also asked for a short-term extension of the current bailout program, Greece’s second, to prevent the impending default. But he offered no substantial concessions that would have won European support, E.U. officials with knowledge of the conversations said.

European officials and international lenders have so far stood firm. Greece’s fiscal situation is widely viewed as a dilemma of its own making after decades of free-spending policies and resistance to belt-tightening.

Tsipras, who was elected in January in an anti-austerity groundswell, has accused creditors of trying to “asphyxiate” the Greek economy, whose debt of more than 300 billion euros is about 180 percent of GDP.

But analysts suggested that the government may be winning more adherents at home than abroad.

“The government here has good cards to play: defiance, resistance, all things that go well with the Greek public,” said Wolfango Piccoli, the managing director of Teneo Intelligence, a consulting firm. But Tuesday’s bailout request was a last-minute tactic, he said.

“The document itself doesn’t look like it was well-thought-out and -prepared,” he said.

U.S. markets took a sanguine view of the situation, with the Dow Jones, S&P 500 and Nasdaq indexes all up slightly Tuesday. European markets were harder hit, but they still did not show the broad sell-off that was a marker of earlier dramas in the long-running Greek crisis.

The missed payment immediately put Greece in what the IMF technically calls “arrears” — and there is no grace period for turning the money in. Under the terms of the loan, Greece is now prohibited from drawing additional funds under any existing arrangement with the IMF.

The consequences could eventually lead to the suspension of Greece’s voting rights within the fund and ultimately its withdrawal.

The IMF does not have the authority to reduce Greece’s debt obligation with the fund, only to waive the requirement for additional payments for up to five years.

Earlier Tuesday, Greek Finance Minister Yanis Varoufakis said he believed that creditors could still find a way to keep the country’s financial lifelines open if they wanted.

“Europe has ways of extending deadlines,” Varoufakis said in an interview.

But Varoufakis also looked ahead to the other possible outcome: A defiant Greece left without its rescue money and under pressure to leave the common euro currency.

Varoufakis said the government could take its case to the European Court of Justice. Greece has “excellent legal grounds” to resist any attempt to oust it from the euro zone, he said, since the agreements that established the euro offer no procedures to leave it.

He said that he believed that if Greeks voted “no” in the referendum, Greek leaders would be strengthened in their tense negotiations with their creditors.

“The referendum is something that we have a duty to do. It’s not a strategy. It’s not a tactic,” Varoufakis said.

Ylan Mui and Brian Murphy in Washington contributed to this report.