Georgia insists that all its forces are now outside South Ossetia
Russia has continued air raids deep inside Georgia, after it rejected Tbilisi's announcement that it had called a ceasefire and wanted talks.
Jets bombed targets near Tbilisi, including the airport, and Russia said its warships had sunk a Georgian boat that approached and tried to attack.
Russia earlier took control of the breakaway region of South Ossetia, forcing Georgian troops to withdraw.
Meanwhile, the US administration has expressed strong support for Georgia.
President George W Bush said that he told Russia's Prime Minister Vladimir Putin: "This violence is unacceptable."
At a meeting on the sidelines of the Olympic in Beijing on Friday, the US president said that all troops should return to their positions prior to the latest unrest, which began on 6 August.
"I was very firm with Vladimir Putin," Mr Bush said.
Meanwhile, Vice President Dick Cheney told Georgian President Mikhail Saakashvili in a telephone conversation that "Russian aggression must not go unanswered".
He said the events in Georgia would have "serious consequences" on Russia's relations with the United States.
Earlier, the US ambassador at the UN accused Russia of seeking "regime change" in Georgia.
President 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.
The airport was hit only a few hours before French Foreign Minister Bernard Kouchner and Finnish Foreign Minister Alexander Stubb arrived on a peace mission.
Meanwhile, there the US clashed with Russia at the United Nations Security Council, accusing it of seeking "regime change" in Georgia.
The US ambassador to the UN, Zalmay Khalilzad, quoted Russia's foreign minister saying Georgian President Mikhail Saakashvili "must go".
He asked his Russian counterpart, Vitaly Churkin: "Is the goal of the Russian Federation to change the leadership of Georgia?"
Mr Churkin did not directly answer the question, but said there were leaders who had "become an obstacle".
The council has met for four days running, but has failed to agree on the wording of a statement calling for a ceasefire.
But the US said it was preparing a draft resolution condemning Russia. Analysts said although Russia would veto such a statement, the US wanted to build backing for the motion to demonstrate international opinion.
Clashes in South Ossetia itself were reported to be less intense on Sunday, as Russian forces took control and Georgian troops drew back.
Mikhail Saakashvili claims Russia has not respected the ceasefire
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 sense of panic on Sunday night in the Georgian town of Gori, near 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.
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
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.
"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.
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