Israel shut down the Palestinian airport in Gaza following an attack on a van of Israeli workers in which seven people were injured, two seriously.
In that event, he would instruct Israeli forces to - as he put it - act accordingly.
Fresh violence broke out on Sunday in the Gaza Strip and West Bank where two Palestinians were reportedly shot dead by Jewish settlers. In northern Israel an Israeli was killed when Palestinians threw a stone through his car window.
Israel shut down the Palestinian airport in Gaza following an attack on a van of Israeli workers in which seven people were injured, two seriously.
Speaking at a news conference, Mr Barak said peace would come one day.
But if Palestinian leaders failed to calm the situation in the West Bank and Gaza, Israel would conclude that Yasser Arafat had abandoned the process of negotiation.
This was a moment of truth for Israel's neighbours, he said.
As Mr Barak was speaking, an Israeli commander announced that a ceasefire had been agreed with Palestinian authorities in the Gaza Strip.
One report said Palestinian police were clearing demonstrators away from a Jewish settlement at Netzarim, one of the flashpoints in the present violence.
UN condemnation
The vast majority of the 83 people killed in the past 10 days have been Palestinian.
The United States - a key ally of Israel - abstained from the vote, but the other 14 members voted in favour.
Political change afoot
On Saturday, Mr Barak said if there was no change in the patterns of violence within 48 hours he would "instruct the Israeli Defence Forces and the security forces to use all means at their disposal to stop the violence".
The Israeli cabinet is to resume meeting to discuss the current crisis; Mr Barak has hinted that he may try to form a government of national unity. A senior Palestinian official rejected the Israeli ultimatum.
Information Minister Yasser Abed Rabbo said the Palestinians would not bow to threats. "If it [the peace process] is dead, it is killed by Mr Barak," he said
Captured soldiers
The crisis deepened on Saturday when the violence spread to Lebanon, where Israel sent its warplanes on the offensive for the first time since withdrawing its forces in May.
Reports say that an Israeli border post has again come under mortar attack in the Shebaa Farms district, which has remained in dispute after the Israeli pullout.
Israel is for the moment said to be concentrating on diplomatic efforts to secure the release of three of its soldiers, who were captured by Hezbollah guerrillas there on Saturday.
Both Lebanon and Syria have been warned by Mr Barak to bring an end to the violence after a bout of cross-border artillery exchanges between Israel and Hezbollah.
President Clinton has telephoned both Mr Barak and Mr Arafat and appealed for calm as part of continuing US efforts to keep any hope of further peace talks alive.