Georgia says its troops have regained control over a strategic gorge in the breakaway Abkhazia region after three days of clashes with a rebel group.
Georgian media disputed the official casualty figures
One woman was killed and two Georgian soldiers wounded in the fighting in the Kodori gorge, Georgian officials said.
But they said militia leader Emzar Kvitsiani - the main target of the Georgian operation - managed to escape.
Abkhaz troops remain on high alert, and Russia has warned Tbilisi it will not tolerate a conflict along its border.
"All villages in the Kodori gorge are liberated, so we can say that the main part of the police operation in the gorge has been completed," Georgy Arveladze, chief of staff of Georgia's president, told reporters.
"Kvitsiani and several of his supporters escaped, and the search for him will be continued. It could last for several days," Mr Arveladze said.
He blamed the death of the woman on the rebels, saying it happened "because Kvitsiani and his people were hiding behind civilians' backs".
Tbilisi said two its soldiers were injured in the clashes, but local media reports said the figure was higher.
Georgia said the insurrection led by Emzar Kvitsiani was backed by Russia.
It accused Moscow of trying to find an excuse for military intervention in Georgia.
Moscow denies any role in the uprising.
Russia has put its troops in the area on alert, condemning the Georgian deployment as a "serious violation of the 1994 Moscow ceasefire agreement", which halted fighting between Georgian and Abkhaz separatist forces.
The unrecognised Abkhaz government also protested against the Georgian move into the gorge - the only part of Abkhazia still under nominal Georgian government control.
Georgia has demanded that Russia withdraw its troops from Abkhazia and another breakaway region - South Ossetia.
Moscow insists that its troops are peacekeepers, needed to prevent a resumption of hostilities.
Abkhaz rebels drove Georgian troops from the region in 1993.