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Swedish Ambassador Olof Skoog called for immediate additional private consultations.
The Security Council was holding those consultations Friday night, but no further vote was expected.
The mandate for the Joint Investigative Mechanism (JIM) to inspect for chemical weapons ran out at midnight on Friday.
Italian UN Ambassador Sebastiano Cardi, President of the Security Council, said the group "will continue to work in the coming hours and days to find a common position in light, of course, of this crucial non-proliferation issue that we have been debating for the last days".
On April 4, more than 80 people were killed in a sarin attack on the rebel-held town of Khan Sheikhun -- an attack that prompted the US to launch 59 Tomahawk cruise missiles at a Syrian airbase, CNN reported.
This is the 11th time that Russia, which supports Assad, has used its veto to stop a resolution on Syria, where some 500,000 people have died in a civil war that erupted in March 2011, reports Efe news.