The US secretary of state also defended on Friday her decision not to meet Hezbollah or Syrian officials when she visits the region soon.
Washington accuses Syria and Iran of financially and morally supporting Hezbollah.
"Syria knows what it needs to do, and Hezbollah is the source of the problem," Rice said before her trip, which begins on Sunday with a stop in Israel.
It is not known if she plans to visit Lebanon.
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 return the two Israeli soldiers whose capture set off the violence 10 days ago and stop firing missiles into Israel.
Rice said: "We do seek an end to the current violence; we seek it urgently. We also seek to address the root causes of that violence.
"A ceasefire would be a false promise if it simply returns us to the status quo."
"Syria knows what it needs to do, and Hezbollah is the source of the problem"
Siding with Israel
The United States has resisted international pressure to pressure its ally Israel to stop the fighting.
The US position has allowed Israel more time to try to destroy Hezbollah in southern Lebanon.
Kofi Annan, the UN secretary-general, also demanded an immediate ceasefire on Thursday and denounced the actions of both Israel and Hezbollah.
Lebanon's prime minister has also asked for an urgent halt to the fighting.
Rice's mission will be the first US diplomatic effort on the ground since the Israeli offensive began.
She said she would visit Israel and the Palestinian territories and join Arab officials, including those from Lebanon, at an international conference on the crisis in Rome next Wednesday.
Italian, French, UN and European Union officials were also expected to attend.
The crisis started on July 12 when Hezbollah captured two Israeli soldiers. Israel retaliated by bombing Lebanon and imposing a naval blockade on the country. Hezbollah fired hundreds of missiles into Israel.
Rice is not expected to try to get a signed deal during her brief visit.
She said: "I know that there are no answers that are easy, nor are there any quick fixes.
"I fully expect that the diplomatic work for peace will be difficult."
The United States is relying on Arab and other intermediaries to put pressure on Hezbollah and Syria.
Washington 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.
As part of a political solution, Rice said there would be a need for a "robust" international force inside Lebanon but said that the US was still discussing with its partners what its mandate would be.
A UN-US plan for long-term stability would give international help to the Lebanese government to install its own troops in southern Lebanon, where Hezbollah exerts control.
"I fully expect that the diplomatic work for peace will be difficult"
Hezbollah "extremists are trying to strangle it in its crib", Rice said of the Lebanese government.
The plan emerged after two days of meetings in New York with Annan and envoys he sent to the region this week.