Russia has warned the US that Kosovo's declaration of independence from Serbia endangers international stability.
Moscow said the comments were made by Russian Foreign Minister Sergei Lavrov during a telephone conversation with his US counterpart Condoleezza Rice.
US President George W Bush earlier defended his move to recognise Kosovo.
In Kosovo on Tuesday, Serbs opposed to independence attacked two border crossings in the Serb-dominated north of the territory, local police said.
At least one border post was burned down. Nato peacekeepers were sent to help police at the Jarinje and Banja crossings.
No casualties were reported among the border officers on duty there, who are reported to have evacuated their posts.
On Monday, Serbia withdrew its envoy to Washington in protest at the US stance. Belgrade says Kosovo's Sunday declaration violates international law.
The UN Security Council is divided over how to respond to Kosovo's move, and it has failed to agree on any action.
Russia and China supported Serbian President Boris Tadic when he made an impassioned appeal to the Council at Tuesday's meeting.
But Britain's representative said that with no prospect of agreement between Belgrade and Pristina the only way forward was supervised independence for Kosovo.
Britain, France, Germany and Italy have all recognised the new state but others have not.
'Threat to stability'
"We confirmed our principled position on the unacceptability of unilateral actions by Pristina declaring its independence," the Russian foreign ministry said in a statement, following talks between Mr Lavrov and Ms Rice.
STANCE ON RECOGNITION
For: Germany, Italy, France, UK, Austria, US, Turkey, Albania, Afghanistan
Against: Russia, Spain, Romania, Slovakia, Cyprus
"We underlined the dangerous consequences of such a step, which threatens the destruction of world order and international stability which have developed over decades," the statement said.
It added that the telephone conversation was initiated by Washington.
The Russian warning came just hours after President Bush said the US would soon establish full diplomatic relations with Kosovo.
Bush backs Kosovo
Speaking during a visit to Tanzania, President Bush said history would prove Kosovo's independence to be "a correct move".
He said there was "a disagreement but we believe as many other nations do that history will prove this to be the correct move".
The president said the US supported Kosovo's independence because "we believe it will bring peace".
In a letter to Kosovo's President Fatmir Sejdiu, Mr Bush offered friendship to Kosovo, and said he supported "your embrace of multi-ethnicity as a principle of good governance".
On Monday, the Serbian parliament passed a resolution condemning Kosovo's declaration of independence.
The resolution also formally annulled the acts of the government in Pristina, saying Belgrade's sovereignty over Kosovo was guaranteed by the UN and international law.
In a separate move, Serbia recalled its ambassadors to the US, France and Turkey because those countries had recognised Kosovo's independence.
Serbian Prime Minister Vojislav Kostunica has threatened to withdraw envoys from other countries which backed the territory's secession.
At a meeting in Brussels, the European Union set aside differences over the recognition of Kosovo's independence, by stressing that the breakaway Serbian province was not a precedent for separatists elsewhere.
All 27 EU foreign ministers agreed to leave recognition up to each member state.
Spain and several other member states have withheld recognition because of concerns about separatist movements within their own borders.
Serbia's interior ministry has filed criminal charges against Kosovo Albanian leaders instrumental in proclaiming independence, accusing them of proclaiming a "false state" on Serbian territory.
HAVE YOUR SAY
Kosovo has declared independence from Serbia, legally or illegally it is not for me to say which
In Belgrade, about 10,000 students marched in protest at the independence declaration, and Serb enclaves inside Kosovo also saw big anti-independence rallies.
In Kosovo's divided city of Mitrovica, three cars were damaged in Tuesday's grenade attack near the office of the Organisation for Security and Co-operation in Europe.
The incident was the latest in a string of hand-grenade attacks on property of international missions in Kosovo and government offices.
Serbian security forces were driven out of Kosovo in 1999 after a Nato bombing campaign aimed at halting the violent repression of ethnic Albanian separatists.
The province has been under UN administration and Nato protection since then.