Georgia insists that all its forces are now outside South Ossetia
Georgia says it has ordered a ceasefire in South Ossetia and offered to hold peace talks with Moscow.
But Russia denied that exchanges of fire had stopped, and continued to bomb targets near Georgia's capital Tbilisi, including the airport, reports said.
Earlier Georgia said its troops had pulled out of the breakaway region and that Russian forces were in control of its capital, Tskhinvali.
Thousands of civilians have fled - it is not clear how many have been killed.
Georgian President Mikhail Saakashvili told the BBC his forces had observed a ceasefire since 0500 on Sunday morning, but had still been bombed by Russian planes. He said his government had been trying "all day" to contact Russia to discuss a ceasefire.
Russian jets were still carrying out bombing raids late on Sunday. Witnesses said jets had hit Tbilisi International Airport, as well as a military airfield close to the Georgian capital.
A Georgian official said Russian planes had also bombed the western town of Zugdidi and Georgian-controlled territory inside Abkhazia. The claims could not be independently verified.
The attack which reportedly hit the airport came only a few hours before the scheduled arrival of French Foreign Minister Bernard Kouchner and Finnish Foreign Minister Alexander Stubb, on a peace mission.
The conflict has caused civilian casualties and more are at risk
Antonio Guterres UN High Commissioner for Refugees
Meanwhile, the United Nations Security Council began meeting for a fourth day on Sunday to discuss the conflict. It has so far failed to agree on the wording of a statement calling for a ceasefire.
Clashes in South Ossetia itself were reported to be less intense on Sunday, as Russian forces took control and Georgian troops drew back.
Local residents fleeing the area on Sunday morning told the BBC that Tskhinvali was relatively quiet.
Later, however, the BBC's Richard Galpin described a real sense of panic on Sunday night in the Georgian town of Gori, near the South Ossetia, amid fears that Russian troops were about to march on the town.
He had been warned by the interior ministry to leave Gori, only to find that the road to Tbilisi was crammed with cars full of fleeing civilians.
Georgia's announcement of its ceasefire came in a statement from the foreign ministry, stating that Georgia "today stopped firing in the South Ossetian conflict zone and is ready to begin talks with Russia on a ceasefire and cessation of hostilities".
It said a note had been passed to the Russian embassy in Georgia to that effect.
Mikhail Saakashvili claims Russia has not respected the ceasefire
But a Russian foreign ministry official was quoted by Interfax saying "our information does not confirm the Georgian statement".
"There are indications that exchanges of fire are continuing and the Georgian forces have not been fully withdrawn from the conflict zone," he said.
The United Nations refugee agency (UNHCR) called on the parties to the conflict to grant safe passage for thousands of civilians trying to escape the war zone.
The UNHCR estimates that between 10,000 and 20,000 people have been displaced within Georgia, including South Ossetia, while Russia has said that a further 30,000 people have fled north into the Russian province of North Ossetia.
ARMED FORCES COMPARED
Total personnel: 26,900
Main battle tanks (T-72): 82
Armoured personnel carriers: 139
Combat aircraft (Su-25): Seven
Heavy artillery pieces (including Grad rocket launchers): 95
Total personnel: 641,000
Main battle tanks (various): 6,717
Armoured personnel carriers: 6,388
Combat aircraft (various): 1,206
Heavy artillery pieces (various): 7,550
Source: Jane's Sentinel Country Risk Assessments
"The conflict has caused civilian casualties and more are at risk," UN High Commissioner for Refugees Antonio Guterres said.
"It is essential that humanitarian agencies be able to reach the affected and the displaced."
Meanwhile tensions were rising in Georgia's other breakaway region, Abkhazia.
The leader of the separatist government there, Sergei Bagapsh, said he had ordered a military operation to clear Georgian forces out of Abkhazia's Kodori Gorge, and gave them a deadline to leave.
Georgia has accused Russia of landing 4,000 more troops in Abkhazia via the Black Sea. The separatists said Georgia had deployed a similar number of soldiers south of the Abkhaz border.
The US has described Russia's actions as "dangerous and disproportionate".
US Deputy National Security Adviser James Jeffrey said that if the Russian escalation continued, it would have a "significant" long-term impact on relations between the Moscow and Washington.
Nato Secretary General Jaap de Hoop Scheffer said Russia had violated Georgia's territorial integrity in South Ossetia and condemned the "disproportionate use of force".
The Russian radio station Echo Moscow reported that two journalists, including a photographer for the Russian news agency Itar-Tass, were shot dead by separatists after entering South Ossetia.
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