Meanwhile, in an exclusive interview with CBS News a top Syrian official denied his country was giving weapons to Hezbollah, and offered to help resolve the crisis.
Prime Minister Ehud Olmert said that Israel will press on with its weeklong Lebanon offensive until captured soldiers are released and Israeli citizens are safe from attacks.
"Israel will continue to combat Hezbollah and will continue to strike targets of the group," Olmert said in a written statement.
Deputy Chief of General Staff Major General Moshe Kaplinsky also told Army Radio the air strikes should be sufficient.
"At this stage we do not think we have to activate massive ground forces into Lebanon but if we have to do this, we will. We are not ruling it out," Kaplinsky said.
In other developments:
Israel struck a Lebanese army base outside Beirut and flattened a house near the border Tuesday, killing at least 16 people in a new wave of bombings. The air strikes are concentrated in the valley in the center of the country and in the south along Lebanon's border with Israel. The Israelis may be aiming for Hezbollah targets, reports CBS News correspondent Elizabeth Palmer in Beirut, but most of the casualties are those who live in Shia Muslim areas where Hezbollah is headquartered and has support.
A new barrage of Hezbollah rocket fire Tuesday afternoon killed one Israeli in Nahariya and wounded several others, Israeli officials said. More rockets also hit the northern city of Haifa, including the train station there, reports — the very same place where a rocket hit two days ago and killed eight people.
A top Israeli defense analyst tells CBS News chief foreign correspondent Lara Logan that the Israelis have not hit all of the strategic targets on their list. All they are looking for now is targets of opportunity, acting on intelligence.
"The one thing they haven't managed to take out is Hezbollah's leadership. That's significant," says Logan. "Here in the Middle East, as long as Hezbollah's leaders stand, they will be seen as heroes for standing up to the most powerful army in the Middle East."
Maj. Gen. Udi Adam, the head of the Israeli army's northern command, warned that the campaign against Hezbelloh wasn't completed.
"It could take days, it could take weeks," he told Israeli Army Radio. "I think that we should assume that it will take a few more weeks."
After a week of fighting, Livni said the time for diplomacy was at hand, though she added that Israel's military operations would not end until its goals are reached.
"The idea is not to reach a ceasefire," Livni said, because that would leave Hezbollah in place to trigger future crises.
Livni's remarks after a meeting with a U.N. delegation to the region were the first indication that the sides were making efforts to end violence that began with a Hezbollah attack on an Israeli border patrol.
"We are beginning a diplomatic process alongside the military operation that will continue," she said. "The diplomatic process is not meant to shorten the window of time of the army's operation but rather is meant to be an extension of it and to prevent a need for future military operations."
Israel was allowing evacuation ships through its blockade of the country. France and Italy moved hundreds of nationals and other Europeans out Monday on a Greek cruise liner. An Italian ship left earlier with 350 people and other governments were organizing pullouts by land to Syria.
"In a situation like this, we have to go out and rent vessels," Burns said. "The problem we have in Lebanon is we are not sure how many of these Americans will come out. We have asked Americans to register with our embassy online; 15,000 have registered, just 3,000 over the last couple of days."
"I think we've made a mistake in policy," said CBS News Military Analyst Ret. Army Col. Mitch Mitchell. "When we see a hot war starting, whether it's going to turn into an all out war or not, we should be doing everything we can to get Americans out and I don't think we're doing a good job at all."
Burns said Secretary of State Condoleezza Rice would be going to the Mideast soon, but couldn't say when.
Progress depends on Hezbollah stopping the shelling of Israel and returning the two soldiers it captured.
"The problem here is Hezbollah and the problem is Hamas starting the fighting over two weeks ago in Gaza," Burns told Early Show co-anchor Hannah Storm. "But there's no question Syria and Iran are arming Hezbollah — they have been doing so for many years."
As a result, Burns said, Hezbollah is now able to reach Israeli cities that haven't seen attacks in 35 years.
"That's due to Iran and Syria providing that missile technology to Hezbollah," Burns said.
Syrian Vice Foreign Minister Fayssal Mekdad told CBS News correspondent Richard Roth the Syrian government is not delivering weapons to Hezbollah, and has not done so since the Syrian army withdrew from Lebanon more than a year ago. Asked if Syria was still giving Hezbollah money and logistical support, Mekdad said, "We don't provide anything."
This is the Syrian government's first official denial since the crisis began.
Mekdad did say Syria provides "support" to Hezbollah, characterized as political and ideological. And in that context, he asserted "We can influence things" in Lebanon. He offered no new formula for ending the crisis. He stuck to the position that a cease-fire has to be the first step, and he dismissed the idea of a release of the captured Israeli soldiers that was not accompanied by a release of Lebanese prisoners held by Israel.
According to the vice foreign minister, the U.S. has made no direct approaches to Syria for help, and he expressed some surprise. "We can help," he said, but don't give them orders. It's clearly a signal Syria wants to get involved diplomatically, though there's no indication it has a new formula to end the crisis.