WASHINGTON (AP) – Secretary of State Condoleezza Rice rejected the “false promise” of an immediate cease-fire in the spreading war between Israel and Hezbollah on Friday and said she would seek long-term peace during a trip to the Mideast beginning Sunday.

The top U.S. diplomat defended her decision not to meet with Hezbollah leaders or their Syrian backers during her visit.

“Syria knows what it needs to do, and Hezbollah is the source of the problem,” Rice said as she previewed her trip, which begins with a stop in Israel.

Rice said the United States is committed to ending the bloodshed, but not before certain conditions are met. The Bush administration has said that Hezbollah must first turn over the two Israeli soldiers whose capture set off the 10-day-old violence, and stop firing missiles into Israel.

“We do seek an end to the current violence, we seek it urgently. We also seek to address the root causes of that violence,” Rice said. “A cease-fire would be a false promise if it simply returns us to the status quo.”

The United States has resisted international pressure to lean on its ally Israel to halt the fighting. The U.S. position has allowed Israel more time to try to destroy what both nations consider a Hezbollah terrorist network in southern Lebanon.

United Nations Secretary-General Kofi Annan demanded an immediate cease-fire Thursday, and denounced the actions of both Israel and Hezbollah. Lebanon’s beleaguered prime minister has also asked for an immediate halt to the fighting.

Daniel Ayalon, Israel’s ambassador to Washington, told The Associated Press that Israel has destroyed about 40 percent of Hezbollah’s military capabilities.

“Most of the long-range (missiles) have been hit, a lot of the medium range, but they still have thousands and thousands of rockets, short-range and others,” Ayalon said in an interview.

He described the Israeli military assault as a “mop up” operation, and said that Israel had no desire to repeat its 18-year occupation of southern Lebanon that ended in 2000.

“They overplayed their hand, they miscalculated,” Ayalon said of Hezbollah militants based in southern Lebanon and supported by Syria and Iran.

Rice’s mission would be the first U.S. diplomatic effort on the ground since the Israeli effort against Lebanon began.

Asked why she didn’t go earlier and engage in quick-hit diplomacy to try to end the death and destruction that has gripped the region, she replied, “I could have gotten on a plane and rushed over and started shuttling and it wouldn’t have been clear what I was shuttling to do.”

The crisis started last week when Hezbollah, an Islamic militant group that operates in southern Lebanon, captured two Israeli soldiers. Israel retaliated by carrying out bombing across Lebanon and slapping a naval blockade on the country. Hezbollah fired hundreds of missiles into Israel.

At least 335 people have been killed in Lebanon in the Israeli campaign, according to the Lebanese health minister. Thirty-four Israelis also have been killed, including 19 soldiers.

Rice plans meetings in Jerusalem and the West Bank with Israeli Prime Minister Ehud Olmert and Palestinian President Mahmoud Abbas, as well as sessions in Rome with representatives of European and moderate Arab governments that are meant to shore up the weak democratic government in Lebanon’s capital Beirut.

Rice’s trip resumes a role the United States has long played as the key Mideast peace broker, but Rice is not expected to try to get a signed deal during her brief visit.

“I know that there are no answers that are easy, nor are there any quick fixes,” Rice said. “I fully expect that the diplomatic work for peace will be difficult.”

The United States is relying on Arab and other intermediaries to pressure Hezbollah and Syria. The United States considers Hezbollah a terrorist group, and has cut high-level ties to Damascus in a dispute over what it says is Syrian meddling in Lebanon.

Hezbollah also exerts political control in southern Lebanon, overshadowing the democratic central government. The U.N. and U.S. plan for long-term stability would give international help to the Beirut government to expel Hezbollah and install its own Army troops, something it has been unable to do.

Hezbollah “extremists are trying to strangle it in its crib,” Rice said of the Lebanese government.

President Bush, asked what he hopes Rice will achieve on her trip, said he would discuss it with her when he returns to the White House on Sunday. He was speaking at a restaurant in Aurora, Colo., as he met with 10 members of the military who recently returned from Iraq.

Announcing plans earlier for a weekend meeting that Bush and Rice will have with Saudi officials, White House spokeswoman Dana Perino said, “This is part of the president’s broader diplomatic outreach on the developing situation in the Middle East.”

Bush and Rice will meet at the White House with Saudi Foreign Minister Saud al-Faisal and Prince Bandar bin Sultan, chief of the Saudi National Security Council.

The plans emerged following two days of meetings in New York with Annan and envoys he sent to the region this week. Annan outlined basic terms of a proposed cease-fire and the longer-range goals to remove the Hezbollah threat in southern Lebanon in a speech on Thursday.