This confidence is rooted in strong signals, largely from Damascus, that Syria may be ready to loosen its ties with Hizbollah, and that the Arab world in general has no sympathy for the guerrillas. According to Israeli intelligence sources, Syria on Friday rejected offers from Tehran to supply Hizbollah with money and arms. This reduced the greatest danger for Israel: an increased Iranian involvement in Hizbollah activity. And early on in the offensive Israel also received indications from Damascus - via United States officials - that Syria would respond only mildly to the bombing and would not intervene. President Assad of Syria remained remarkably silent, despite the civilian suffering caused by the Israeli bombardment.
As the week wore on, therefore, Israel came to believe that a 'green light' for the offensive had been given, and that Damascus would eventually help to engineer a ceasefire, after Hizbollah had been weakened.
It remains unclear, however, whether Hizbollah will agree to end its rocket attacks on Israel. Also uncertain is whether the US, which must now broker a formal end to the conflict, can pay the price that Syria is certain to demand for supporting a ceasefire. Shia Hizbollah gunmen operating in south Lebanon are trained and armed largely by Iran. Syria lends them tacit support, allowing arms to flow through Damascus and encouraging their activities in Lebanon. Israel's main stipulation for a ceasefire was that Syria should curb Hizbollah, which has increased its attacks against Israeli targets in recent months.
Israeli government sources and Western diplomats in the Middle East believe Syria was signalling that it would be flexible to such demands, and that the US had tacitly agreed to the Israeli action. 'There is a sense in which all of this was pre-ordained. Otherwise why has the world remained so quiet?' said a Western official.
While attention has focused on the bombing in Lebanon, frenetic diplomatic activity has been underway all week. The first sign that Syria was adopting a flexible approach came when Damascus did not condemn the deaths of four of its soldiers in the bombardment last Sunday. US contacts were immediately established with Syria and Israel; and Bill Clinton praised Syria for its 'restraint'. Lebanon, which called for Security Council action last Sunday, withdrew its request on the bidding of Syria. And on Thursday, Iranian Foreign Minister Ali Akbar Velayati travelled to Damascus, offering money and arms for Hizbollah.
Now Israel would like the US to consummate diplomatically what it believes it has achieved militarily, by offering Syria incentives to cut off Hizbollah.
This is a lot to ask, even of the US. Warren Christopher, the US Secretary of State, is due in the region on Tuesday. But as one Western diplomat put it yesterday: 'I'm not sure exactly what we have to offer them (the Syrians).'
Mr Assad is already angry that Syria has not been better rewarded by the US for its support in the Gulf war. The Syrian President would like his country to be removed from the US list of states sponsoring terrorism. But Mr Clinton's hands are tied by Congress, which takes a strongly anti- Syrian position.
The only viable forum for offering Syrian incentives to ensure peace in southern Lebanon is in the context of the peace talks. Syria is now sure to redouble is demands for the return of the Golan Heights, as a quid pro quo for curbing Hizbollah.
However, the conflict of the past week may well have made it even more difficult for Yitzhak Rabin, the Israeli Prime Minister, to return the Golan to Syria. For the Israeli public, this week's fighting has proved the need for more security on the state's borders, not less.