U.S. Secretary of State Condoleezza Rice said the need for a cease-fire between Israel and Hezbollah fighters is "urgent," but only when the conditions are right. Rice spoke with reporters en route to the Middle East, where she is scheduled to meet with Israeli officials Monday evening. "We believe that a cease-fire is urgent," Rice told reporters on a flight from Washington to a refueling stop in Ireland. "It is important, however, to have conditions that will make it sustainable." Rice's visit follows trips to the region by European and U.N. diplomats who joined Lebanon's calls for a cease-fire. The United States has not called for an end to the fighting, arguing that leaving Hezbollah in place on Israel's northern border would only make further conflict inevitable. Rice said she has been consulting with U.N. and Israeli officials about elements of a cease-fire that would ensure Lebanon had control of its country. "The really important thing here is that whatever we do has to contribute to Lebanon's regaining sovereignty over all its territory," said Rice. "It's just very important that we work urgently, but that we also work in a way that is going to push this forward, not backwards." Rice is scheduled to visit Israeli Prime Minister Ehud Olmert in Israel and Palestinian President Mahmoud Abbas in the West Bank. She will also attend an international conference in Rome, Italy, on crafting a peace agreement and shoring up Lebanon's government, CNN reported. Marking the level of diplomatic leverage being applied to this trip, U.S. Assistant Secretary for Near Eastern Affairs David Welch and deputy national security advisor Elliot Abrams are traveling with Rice and will stay in the region as the secretary of state continues to Rome and Kuala Lumpur later in the week. "We intend to treat the government of Lebanon with the respect that it deserves and also with the great desire ... to see it able to extend its sovereignty over its territory," Rice said.