Pro-Russian insurgents in eastern Ukraine agreed Monday to respect a ceasefire declared by the Ukrainian president, raising hopes for an end to months of fighting that have killed hundreds and ravaged the country's industrial heartland.

The announcement came as the Russian and U.S. presidents traded demands over the conflict. Russian President Vladimir Putin urged direct talks between the government and the rebels. U.S. President Barack Obama warned Putin that Moscow will face additional costs if it does not help ease the crisis.

The insurgents' pledge to respect the ceasefire came on the first day of talks held in Donetsk between a former Ukrainian president, the Russian ambassador, European officials and the eastern separatists who have declared independence.

Ukraine’s government side was represented by ex-President Leonid Kuchma, who attended the discussions at the request of sitting president Petro Poroshenko (Poroshenko has ruled out talks with those he calls "terrorists," so inviting Kuchma to mediate offered a way to conduct talks without the government's formal engagement.)

The negotiations were launched in line with Poroshenko's peace plan, which started Friday with a weeklong unilateral ceasefire in the fighting that has killed more than 350 people and forced tens of thousands to flee their homes.

Alexander Borodai, one of the rebel leaders who took part in Monday's talks in Donetsk, said rebels would respect Poroshenko's ceasefire, which lasts until Friday.

The insurgents had previously demanded the Ukrainian military withdraw its troops from the east as a condition for any talks, so Borodai's statement represented a softened stance that raised expectations that the ceasefire could hold. Even before the insurgents made their pledge, the government said that there had been no fighting in the east since Monday morning.

Ceasefire hasn't stopped violence so far

Since the ceasefire was declared Friday, the Ukrainian government has accused the rebels of firing at government positions, while insurgents blamed Ukrainian forces for failing to honor the promise to halt hostilities. Poroshenko has said government troops will fire back if attacked.

Russia Vladimir Putin

Russian President Vladimir Putin said it was necessary to "end fighting and start direct talks" between the rebels and Ukraine's government to resolve the situation in eastern Ukraine. (Maxim Zmeyev/Reuters)

The rebels, who have declared regions on border with Russia independent and fought government troops for two months, also promised to release observers from the Organization for Security and Cooperation in Europe who have been held hostage.

"This will be one of the steps that will improve the mutual understanding of both sides," said Alexei Karyakin, a representative of the insurgents in the Luhansk region.

In Moscow, the Kremlin said Putin underlined in his conversation with Obama that to normalize the situation in eastern Ukraine, it's necessary to "effectively end fighting and start direct talks between the conflicting parties."

White House spokesman Josh Earnest said Obama urged Putin to use his personal influence with the separatists to promote peace and stability in Ukraine, stop backing the insurgents and halt the flow of arms across the border.

Earnest said that while the U.S. believes a diplomatic solution to the crisis is still possible, "Russia will face additional costs if we do not see concrete actions to de-escalate the situation."

Ukraine and the West have accused Russia of fomenting the rebellion in the east by sending troops and weapons across the border. Moscow has denied that and insisted that Russian citizens who joined the insurgents were volunteers.