BRUSSELS, Belgium -- President Barack Obama and British Prime Minister David Cameron laid down a new deadline for Russia on Thursday, giving Moscow a month to meet their conditions in Ukraine or face further sanctions.

Russian President Vladimir Putin, the focus of intensifying diplomatic activity this week, didn't immediately respond to the new conditions. Sustained violence in eastern Ukraine is driving a frenzy of high-level meetings in Brussels, Paris and Normandy aimed at reconciling Russia and Ukraine.

Putin is holding his first face-to-face meetings with Western leaders: Cameron and French President Francois Hollande on Thursday night, and German Chancellor Angela Merkel on Friday. There's hope that Putin could also talk with Ukraine's new president-elect.

Obama and Cameron spelled out new demands for Russia at a joint news conference after a Group of Seven world leader summit. Putin was meant to have hosted the summit in Sochi, but the G-7 countries cancelled that after his aggressive moves in Ukraine, and met without him in Brussels.

It was the group's first summit in two decades without the participation of Russia.

The U.S. and Europe, after imposing economic sanctions on Russia in recent months, are considering toughening them.

To avoid even harsher sanctions, Cameron said Putin must meet three conditions: recognize Petro Poroshenko's election as the new leader in Kyiv, stop arms from crossing the border and cease support for pro-Russian separatist groups concentrated in eastern Ukraine.

"If these things don't happen, then sectoral sanctions will follow," Cameron said. "The next month will be vital in judging if President Putin has taken these steps."

After meeting Putin on Thursday night, Cameron said he sent a "very clear and firm set of messages" about the crisis in Ukraine and said the status quo was "not acceptable."

Obama said the G-7 leaders unanimously agree with the steps Cameron outlined. But they weren't so explicit in written statements issued after two days of meetings, and an Obama aide later described the potential sanctions in different terms than Cameron.

"If Mr. Putin takes those steps, then it is possible for us to begin to rebuild trust between Russia and its neighbours and Europe," Obama said. "We will have a chance to see what Mr. Putin does over the next two, three, four weeks, and if he remains on the current course, then we've already indicated that kinds of actions that we're prepared to take."

Obama acknowledged that so-called sectoral sanctions, which would hit key sectors of Russia's economy, could have a bigger impact across Europe because of European economic ties to Russia, and said he didn't necessarily expect all European countries to agree on them. But, he said, "it's important to take individual countries' sensitivities in mind and make sure that everybody is ponying up."

"My hope is, is that we don't have to exercise them because Mr. Putin's made some better decisions," Obama said.

Putin may claim that he has already fulfilled the conditions. He has already welcomed the Ukrainian election and said Russia would "respect the will of the Ukrainian people." On sealing the border and withdrawing support to separatists, he insisted in a French television interview this week that the U.S. has "no proof" that Russians are playing any role at all in the unrest in eastern Ukraine.

Russia has also said it will send its ambassador to Ukraine for Poroshenko's inauguration. Ambassador Mikhail Zurabov had been recalled a few days after street protests pushed pro-Moscow President Viktor Yanukovych to flee in February.

Obama said he thought the fact Putin didn't immediately denounce the outcome of Ukraine's election last month offers hope he's moving in a different direction. "But I think we have to see what he does and not what he says," he added.

The G-7 countries expressed their opposition Thursday to Russia's annexation of the Crimean Peninsula, but Obama and Cameron notably didn't mention Crimea when laying out their conditions for Moscow to avoid new sanctions.

It's not clear whether that means the G-7 is quietly accepting Russia's de facto control of the peninsula. U.S. officials have long said privately they didn't expect Russia to give up Crimea now.

While Putin was pointedly excluded from the G-7 summit, some leaders seemed to show an openness to allowing him back into the international fold after months of isolation.

There's one key leader Putin is not scheduled to meet this week -- Obama. Amid the worst East-West crisis in a generation, U.S.-Russia relations are so tense that the French president is eating back-to-back dinners Thursday night so that Obama and Putin don't have to share the same table.

Obama said they may have an opportunity to talk when they both attend events commemorating 70 years since the D-Day Allied invasion that helped wrest Western Europe from Hitler's grip.

Putin's meetings with Cameron, Hollande and another with Merkel on Friday illustrate how Western countries are divided over how to deal with the Russian leader.

France's president said that Putin could meet with Poroshenko when both come to Normandy. European leaders are hoping the famous beaches may now serve as a diplomatic platform to end the Ukraine-Russia crisis.

"It is an exceptional international meeting that must serve the cause of peace," Hollande said.

The top U.S. and Russian diplomats met in Paris on Thursday, and agreed that Ukraine should be a bridge between Russia and the West and not be a "pawn" in a power struggle.

But Russian Foreign Minister Sergey Lavrov, after meeting with U.S. Secretary of State John Kerry, also accused the U.S., NATO and G-7 of aggravating tensions in Ukraine. "No outside players, influencing different Ukrainian parties, should ignite passions," he said.

The May 25 election of Poroshenko, a billionaire candy tycoon, was seen as a critical step toward resolving Ukraine's protracted crisis.

Since February, Russia has annexed the Crimean Peninsula in southern Ukraine, the eastern regions of Luhansk and Donetsk have declared their independence from Kyiv, and the interim Ukrainian government has launched an offensive in the east to quash an uprising that has left dozens dead.

AP writers Vladimir Isachenkov in Moscow, Angela Charlton and Matthew Lee in Paris, and Juergen Baetz, John-Thor Dahlburg and Julie Pace in Brussels, contributed to this story.