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Russian soldiers acting as peacekeepers patrol areas between Georgian and Abkhazian forces but handing full military control of the breakaway province to the Kremlin would alarm both the Georgian government and its allies in the West.
"Those 200 km (120 miles), the distance between the Psou and the Inguri rivers, are all Abkhazia. We agree to Russia taking this territory under its military control," Sergei Shamba, foreign minister of Abkhazia, told the Russian newspaper Izvestia.
"In exchange, we will demand guarantees of our security."
Russian Foreign Minister Sergei Lavrov said Moscow had not received an official request from Abkhazia for its military to take control of the region.
After the NATO summit, Moscow announced plans to establish legal links with Abkhazia and South Ossetia, another "frozen conflict" region inside Georgia.
NATO has urged Russia to reverse the steps and complained that the deployment of extra troops would add to tensions. The European Union has also expressed concerns.
Lakobashvili said Georgia was urging the European Union to take a more active role in reducing tensions, with options including participating in border control or policing.
"We should have more Europe in these conflict zones," he said, while adding that no decisions on a bigger EU role had been taken during his talks in Brussels.