four and a half miles inside to engage Hezbollah fighters and destroy their outposts and infrastructure.

The Lebanese news media reported that at least one helicopter with Israeli commandos landed near Baalbek, in eastern Lebanon, a Hezbollah stronghold well north of the Litani River, marking a deeper, if limited, invasion. The Hezbollah television station Al Manar said its fighters repelled the Israeli troops. Israeli officials declined comment.

Israel continued its promised 48-hour "partial suspension of aerial activity," as the army called it, but there were numerous sorties. The air force flew missions in support of ground troops and to hit Hezbollah targets that included, the army said, two groups of fighters who were launching rockets, missile launchers, missile launching sites, "access routes" in the Bekaa Valley that were used to bring weapons from Syria, and "Hezbollah structures and headquarters."

At least one truck suspected of carrying weapons was bombed near the Syrian border, the Israeli Army said. Lebanese said there were repeated airstrikes in the area, where the highway from Damascus crosses the Bekaa, especially in the Shiite area of the valley known as the Hermil, where one pickup truck carrying canisters of cooking gas was attacked with rockets.

In northern Israel, red-and white tourist buses arrived along the border filled with soldiers who had been waiting weeks for orders. They moved into Lebanon through corridors cleared by bulldozers, tanks and engineering units.

There were house-to-house battles with hundreds of Hezbollah fighters in Lebanese towns and villages close to the border, especially around Aita al-Shaab, northeast of Shtula, where Hezbollah fighters breached the border on July 12, capturing two Israeli soldiers and killing eight, igniting this 20-day-old conflict.

There were other battles to the west around Al Taibe and Maroun al-Ras, Adessa and Rab-e-Talatin, the Israeli Army said.

At least 3 Israeli soldiers and more than 20 Hezbollah fighters were killed in periods of heavy fighting, according to Arabic television and the Israeli Army.

Arab satellite television stations broadcast live images of smoke rising from the villages and there was the sound of heavy machine-gun fire.

Israeli troops may push northward to the Litani River, 15 miles from the border, cabinet ministers said following their meeting, which ended in the early hours on Tuesday. But the Israeli intention now seems to be to clear out a wide strip of land along the border into which an international force could deploy without itself having to fight Hezbollah, a cabinet minister said.

Israeli military officers said they suspected they would have a limited time to pursue their objectives - perhaps a week or so more - and were trying to map out their final goal.

If an international force is long delayed or does not materialize, officials said, Israel is likely to proceed to the Litani River, which marked the southern Lebanon "security zone" that Israel left in 2000.

Prime Minister Ehud Olmert said Tuesday evening, "We are at the beginning of a political process that in the end will bring a cease-fire under entirely different conditions than before."

Speaking at the graduation ceremony at the National Security College, Mr. Olmert said, "The State of Israel is winning in this battle, and is gaining impressive achievements, perhaps unprecedented ones."

He added: "If the military campaign would have ended today, today we could already say with certainty the face of the Middle East has changed."

Mr. Olmert's conduct of the war has been criticized by some Israelis as being too slow or timid, but he defended himself.

"This threat will not be what it was," he said of Hezbollah. "Never will they be able to threaten this people they fired missiles at. This people will defeat them."

But he continued to reject calls from various countries, including Arab ones, for an immediate stop to the fighting. "Every additional day is one that erodes the power of this cruel enemy," Mr. Olmert said.

Meeting in Brussels, the European Union foreign ministers called for "an immediate cessation of hostilities, to be followed by a sustainable cease-fire." The ministers called for the United Nations Security Council "to be rapidly convened to define a political framework for a lasting solution agreed by all parties, which is a necessary precondition for deployment of an international force," to which, they said, some European countries would provide troops if the rules of engagement were right.

But Mr. Olmert and the Israelis say they will not stop fighting until a political package is in place, including the release of captured soldiers and the formation of a buffer force that can take the place of the Israeli Army inside Lebanon without leaving a power vacuum that Hezbollah could use to resupply or re-infiltrate the area.

"A few of our forces are already along the Red Line of the former security zone and some are already beyond it," said Brig. Gen. Shuki Shachar, deputy commander of the northern command, referring to the Litani River. Other officers said that Israeli forces north of the Litani were commandos, not infantry units fighting in mass.

"We have so far now about six efforts running inside Lebanon, a kind of brigade size or bigger than a brigade in each one, all of them mainly infantry," the general told reporters during a briefing at command headquarters in Safed.

An Israeli brigade can contain anywhere from 900 to 1,600 soldiers. Other officers of lower rank said the total number of soldiers now inside Lebanon was 5,000 to 7,000 and was expected to climb higher.

"The ground campaign is becoming bigger and bigger from day to day," General Shachar said.

In the Lebanese border town of Kafr Kila, where artillery fire was heavy, residents said Israeli tanks had approached but had been forced back. The Israeli Army said it had no reports of tank fighting around the village, though nearby villages were attacked.

Hezbollah fighters were seen replenishing stocks of ammunition and preparing positions, journalists inside Lebanon reported, adding the fighters expected a new Israeli push on Wednesday after 2 a.m., when the 48-hour Israeli aerial pause expires.

General Shachar said Israel already controlled the Litani River with air power or artillery in some places and in others, such as where the river turns close to the border, with troops on the ground. He said the military had left two bridges over the Litani intact - one in the east near Hasbayah and another in the west north of Tyre - to allow people to move north and humanitarian aid to move south.

The growing Israeli offensive is targeting a relatively small enemy. Israeli military officers estimate Hezbollah's active fighting force at 2,000 to 3,000 men, 250 to 300 of whom Israel says it has killed.

"It isn't easy to accomplish the mission, because they are in small teams and are spread over such great distances," General Shachar said.

"It's bigger, it's increased in its scope in numbers of troops and in area," said a senior military officer. "But the mission is still the same, as in Bint Jbail and Maroun al-Ras," which is to clear Hezbollah fighters, explosives, mines, outposts, storage areas, barracks and other infrastructure from a band of territory two to three miles wide along the border, into which an international force can be inserted without having to fight Hezbollah.

Last week, the cabinet called up some 30,000 reserve soldiers, many of whom reported to their bases earlier this week to begin training. The reserves will be used partly for Lebanon and partly to be ready in case Syria, whose military is on high alert, chooses to broaden the war, Israeli officials say.

"We have reached the stage where we have to expand the operation," said Defense Minister Amir Peretz, without giving the dimensions of the next phase.

"The goal is to hit at Hezbollah fighters and their weapons arsenals," Justice Minister Haim Ramon said in a televised interview. "And today, we are not doing a bad job."

As the world talked about the new multinational force to go into Lebanon, its makeup and rules of engagement were vague. There is also the murkiness of Lebanese politics, sectarian-based, and the meddling influence of Hezbollah's main sponsors, Syria and Iran.

The central contradiction is that, while Israel and the United States regard Hezbollah as a terrorist organization, most Arabs and many, although not all, Lebanese view them as heroes who forced Israel out of southern Lebanon in 2000, after an 18-year occupation.

The last few weeks have essentially transformed Lebanese politics, marginalizing the democratic forces promoted by the United States and France - known as the March 14 group - and instead empowering President Emil Lahoud, a staunch ally of Syria, and, above all, the Shiite Muslim speaker of parliament, Nabih Berri, who is the only official link to the Hezbollah leader, Sheik Hassan Nasrallah.

France's defense minister, Michèle Alliot-Marie, said in an interview published in Paris on Tuesday that any international force should have 15,000 to 20,000 troops, far larger than the current United Nations force posted there, and have rules of engagement that would permit its soldiers to open fire when necessary.

"It must be credible and capable of making itself respected by everyone," Ms. Alliot-Marie said. France is considered likely to take the lead in providing a core of any force sent to southern Lebanon.

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Craig S. Smith reported from Misgav Am, Israel, for this article, and Steven Erlanger from Jerusalem. John Kifner contributed reporting from Beirut, Lebanon, and David Stout from Washington.