French Ambassador Jean-Marc de la Sabliere said discussions would now be renewed outside the UN.
Russia has vowed to veto a draft resolution that would give Kosovo "supervised independence" - based on recommendations of a UN envoy.
Kosovo - currently part of Serbia - has been governed by the UN since 1999.
The province's ethnic Albanian majority wants independence but its Serbian minority - and the government in Belgrade - opposes this.
Russia has promised to represent the views of its ally, Serbia, at the UN Security Council, where it holds the power to veto resolutions.
Serbian Prime Minister Vojislav Kostunica said the decision to put on hold plans to push through Kosovo's independence at the UN was an "important victory" for Belgrade and Moscow.
However he insisted that Belgrade was still prepared to engage in discussions on the province's future.
The French ambassador to the UN said on Friday that it had been impossible to reach agreement on a draft resolution in the Security Council.
We will therefore put on hold discussions on the resolution," Mr de la Sabliere said.
Güncelleme Tarihi: 21 Temmuz 2007, 11:47
France is currently holding the rotating presidency of the Security Council.
Mr de la Sabliere said an agreement on Kosovo would now be pursued through the Contact Group - a body consisting of the US, France, Germany, the UK, Italy and Russia.
He added that the issue had to be resolved "as soon as possible" and that the UN plan remained the best way forward.
No member of the Contact Group has the right to exercise a veto on decisions.
Asked why the draft resolution was not put to a Security Council vote, UK envoy to the UN Emyr Jones Parry, said: "We don't want high drama which might have a consequence in the region. We have got to be responsible."
Kosovo came under UN administration after the withdrawal of Serbian forces in 1999.
The Serbian pull-out came after Nato air raids, launched to halt a Serb crackdown on ethnic Albanians, some of whom had taken up arms.