Ehud Olmert said there would be no stopping in the two-week offensive and that “severe measures” would be taken against Hezbollah.
Speaking in Jerusalem ahead of the meeting he said, “Israel is determined to continue on in the fight against Hezbollah. We will … stop them.
“We are using the basic elementary right of self-defence.”
Rice said a ceasefire was needed in the region, but not at any price.
“A durable solution will be one that strengthens the forces of peace and democracy in the region,” Rice said before the talks.
“It is time for a new Middle East. It is time to say to those who do not want a new Middle East that we will prevail,” she added.
Rice told Lebanese leaders on Monday that Hezbollah must return two captured Israeli soldiers and withdraw from Israel’s border before there could be a ceasefire, Lebanese politicians said.
“I have no desire to be back in three weeks or three months or six months when once again extremists have decided to use their advantages to destabilise the region,” she said.
Washington is arguing that UN resolution 1559 and the Taif Agreement, which ended the Lebanese civil war in 1990, need to be fulfilled.
Both documents call for the Lebanese government to exercise full control over its territory, and the disarmament of militias -including Hezbollah.
Lebanese politicians want an immediate ceasefire before a long term deal, but Israel wants Hezbollah to leave the border area and free the captured soldiers without conditions.
Ground raids and air strikes have failed to stop around 1,200 rockets being fired into northern Israeli towns and cities, where they have killed 17 civilians so far.
Rice left Israel after the talks to meet Mahmoud Abbas, the Palestinian president, before she heads to an international conference in Rome on the conflict on Wednesday.
Rice assured Abbas that the US has not forgotten the Palestinians’ plight and discussed getting additional aid to the debt-laden Palestinian government.
During the meeting that included about a dozen US and Palestinian officials, Rice and Abbas talked about the state of the Israeli soldier captured last month by Palestinian gunmen.
Rice told Abbas that the US had
Rice said she briefed Abbas “on efforts we’re making to bring about an urgent but enduring cease-fire in Lebanon, one that can deal with the causes of extremism that began this crisis and that can also lead to the establishment of the sovereignty of the Lebanese government throughout its territory”.
Rice vowed that the US would not tire in its attempts to achieve “two states living side by side in peace”.
Abbas renewed a call for an Israeli-Palestinian truce, following a month-long Israeli offensive in the Gaza Strip, launched to free a captured Israeli soldier.
“We are exerting all our efforts to release the Israeli soldier,” he said, adding that he hoped thousands of Palestinian prisoners would also be freed by Israel.
“Israeli aggression in the West Bank and Gaza Strip must stop immediately so we can strengthen the truce and start a political process that aims to end the occupation,” he said.
More than 2,000 Palestinians took to the streets of Ramallah before Rice’s visit to show support for Hezbollah and protest against US policies.
The protesters walked through the centre of the capital, where most of the shops closed down after calls for a general strike, shouting “Bush is a criminal”, “The United States is a terrorist” and “Rice, Rice, you are a crow, what misery you bring with you”.
Stone-throwing demonstrators also tried to break into Abbas’ headquarters, but were stopped by security forces there.
European and Arab foreign ministers are set to discuss proposals for an international “buffer” force of the border between Israel and Lebanon at the Rome meeting.
Several European Union nations have said they were ready to contribute to the force.
Amir Peretz, the Israeli defence minister, said on Tuesday that Israel would maintain a security zone until the proposed international force arrives there.
Peretz did not say whether soldiers would stay in Lebanon or would maintain the zone using air strikes and artillery fire.
“There will be a security zone, which will be under the control of our forces if there is not a multinational force,” he said. “If there is not a multinational force that will get in to control the fences, a multinational force with an enforcement capability, we will continue to control (Hezbollah) with our fire toward any one who will get close to the defined security zone.”
About 750,000 people have been
The US backs the idea of a humanitarian corridor in Lebanon to get help to the needy, an idea Israel says it could support.
At least 408 people in Lebanon and 41 Israelis have been killed in the violence which began after Hezbollah seized two Israeli soldiers in a cross-border raid on July 12.
About 750,000 Lebanese have been displaced.
Israel’s offensive in the Gaza Strip is also continuing to try to recover a soldier captured by Palestinian fighters and halt rocket fire.
Israeli forces have killed at least 121 Palestinians in the month since then.