The new draft was presented by Great Britain on behalf of the United States and Europe.
"The process to move forward has started, Kosovo independence is inevitable," said US Ambassador Zalmay Khalilzad.
The most significant change in the revised draft is a regulation "supporting" Kosovo's independence, while the previous draft "approved" independence.
Asked by reporters if the countries that proposed a new draft wold stick to it despite the threat of Russia's veto, Khalilzad said they had acknowledged the Russian veto and that a draft resolution currently under discussion would be discussed to try to meet Russian demands.
"We'll make every efforts to accommodate the constructive ideas from our Russian colleagues," Khalilzad told reporters, adding however that the balls were currently in Russia's court..
Spokesman for the State Department Tom Casey said on Thursday that there was a consensus that the time had come to conclude the Kosovo issue in accordance with the fundamental guidelines of Martti Ahtisaari's plan. He said the United States wanted everyone to support the resolution, adding that this was important for the stability of the region, people in Serbia, Kosovars and others in the region.
The draft calls for the "full implementation" of a UN comprehensive proposal to make Kosovo an independent territory under international supervision involving the European Union and the Organisation for Security and Cooperation in Europe.
An international representative would be appointed to oversee the independence process.
The comprehensive proposal was drawn up by former Finnish president Martti Ahtisaari and emphasizes the importance of
establishing a "multiethnic Kosovo".
Moscow and Belgrade have opposed the plan and demanded more negotiations between Belgrade and Pristina.
Under the draft resolution, NATO, which has been providing security since 1999, will remain in Kosovo.
Kosovo has been administered by the UN in cooperation with other European missions and NATO since 1999, following the military
intervention by NATO to expel Serb security forces accused of repressing the ethnic Albanian majority.