Russia was pointedly disinvited from a gathering Thursday of the exclusive Group of Seven nations, but it was clear that the international big chill imposed on Moscow over its actions in Ukraine is starting to thaw.

President Obama and other leaders meeting here offered an olive branch to Russian President Vladimir Putin, inviting him to begin to resolve the seven-month Ukrainian crisis and rebuild trust with the West after an estrangement that invited comparisons to the Cold War.

Putin has an opportunity “to seize this moment” and turn the page, Obama said.

Putin should recognize that Ukraine’s newly elected president, Petro Poroshenko, is the “legitimate leader of Ukraine,” he said. Obama also said he was pleased to see Putin not denounce Poroshenko’s May 25 election, adding that it “offers the prospect that he’s moving in a new direction.”

Although the Group of Seven warned that Russia would face harsher sanctions if it fails to ease months of tension, new economic penalties are on hold, to the evident relief of European nations.

The detente with Russia continued with scheduled meetings between Putin and the British and French leaders Thursday in Paris. German Chancellor Angela Merkel is expected to see Putin on Friday. Only Obama and Canadian Prime Minister Stephen Harper refused to meet with Putin one on one when the allies of World War II gather in France to commemorate the 70th anniversary of D-Day.

Putin’s Russia was pushed out of what had been known as the Group of Eight over the invasion and annexation of Crimea this spring. The gathering of heads of state was abruptly moved from Sochi, Russia, to the Belgian capital, home to the European Union and NATO.

British Prime Minister David Cameron said Thursday that Russia had lost its seat in the economic club by actions that are “totally at odds with the values of this group of democracies.” He went on to offer what appeared to be specific criteria for avoiding stricter sanctions: the end of Russian support for separatists operating in eastern Ukraine and the recognition of the Poroshenko government.

“If these things don’t happen, then sectoral sanctions will follow,” Cameron said, referring to much tougher penalties targeting sectors of the Russian economy.

Modified sectoral sanctions, largely targeting the Russian defense industry, were readied in May in anticipation of potentially heavy Russian interference in the Ukrainian elections, which Russia had called illegitimate. Those sanctions were considered a last resort by many European nations that trade heavily with Russia. Moscow surprised and pleased European nations by refraining from overt meddling in the election, and European diplomats said the strong hope now is that sectoral sanctions will quickly become unnecessary.

Obama and Cameron suggested that Putin has roughly until the end of the month to decide whether he will change course in Ukraine.

“If he continues a strategy of undermining the sovereignty of Ukraine, then we have no choice but to respond,” Obama said at a news conference with Cameron, who expressed the same view.

“Sectoral sanctions are broader; they’d be more significant,” Obama said. “My hope is, is that we don’t have to exercise them because Mr. Putin has made some better decisions.”

The comments by Obama and Cameron were striking because they suggested that the leaders would seek to impose harsher sanctions on Russia if Putin does not take active steps to resolve the crisis in Ukraine.

Many in Eastern Europe have worried that the West would not impose additional punishments if the current situation continues in eastern Ukraine, where separatists widely believed to be backed by Russia are stirring unrest and violence.

“I think Europeans understand that the reason we’ve seen such extraordinary growth and peace on this continent has to do with certain values and certain principles that have to be upheld,” Obama said. “And when they are so blatantly disregarded, the choice is clear: Europeans have to stand up for those ideals and principles even if it creates some economic inconvenience.”

A joint statement from the G-7 condemned what it called Russia’s illegal annexation and “unacceptable interference” in other areas of Ukraine and said the body will refuse to recognize Crimea as a part of Russia.

“We stand ready to intensify targeted sanctions and to implement significant additional restrictive measures to impose further costs on Russia should events so require,” the statement said.

U.S. officials later said certain provocative actions — such as Russian troops crossing the border into Ukraine — would trigger more punitive measures. But they would not define exactly what Putin would have to do, nor would they set an exact deadline.

“Clearly sectoral sanctions are on the table,” said Ben Rhodes, the deputy national security adviser for strategic communications. “If we see the status quo continuing, we are going to move.”

Obama and Cameron acknowledged that there are differences of opinion among the 28 European Union nations and that rallying them all to favor stricter sanctions would be difficult.

“Do I expect unanimity among the 28 E.U. members?” Obama joked.

“Welcome to the club,” Cameron interjected with a laugh.

Cameron said his meeting with Putin was a reasonable way to keep the lines of communication open to Moscow.

U.S. officials have been clear that they would have preferred a unified rebuff to Putin this week, although Obama said it is entirely appropriate for Putin to attend D-Day commemorations in Normandy on Friday. “I have no doubt that I’ll see Mr. Putin” there, he said, “and should we have the opportunity to talk, I will be repeating the same message that I’ve been delivering to him throughout this crisis.”

While Obama would not meet with Putin, Secretary of State John F. Kerry met with Russian Foreign Minister Sergei Lavrov in Paris on Thursday. Kerry repeated that he hopes Russia will validate and meet with the new Ukrainian government. The goal, he said, is for Ukraine not to be a pawn.

“The Russian-American agenda is much broader than just Ukraine,” Lavrov said. “We would like to see other countries like Iraq, Libya, Syria, many others, also to be in peace, not to be used as a pawn, and I hope that we can discuss all these things.”

As the leaders talked in Brussels, civilians were evacuating the eastern Ukrainian regions where the Ukrainian military has been engaged in ongoing clashes with pro-Russian separatists.

In the past 24 hours, more than 8,300 Ukrainians, many of them children, have sought refuge in Russia’s southern Rostov region, said the region’s governor, Vasily Golubev, according to the Russian Interfax news agency. About 4,000 Ukrainians have applied for refugee status in Russia, Prime Minister Dmitry Medvedev said at a government session Thursday.

Ukrainian military officials said there had been several sporadic am­bushes on troops deployed in the east. The Ukrainian Border Guard Service said insurgents fired Thursday on guards and soldiers manning a checkpoint near the Russian border. The rebels’ vehicle and a machine gun were destroyed, the border guard said.

Gearan reported from Washington. Carol Morello contributed to this report from Kiev.