Even as Israel's leadership denied that a deal was reached with Hamas on a cease-fire, Palestinian Authority President Mahmoud Abbas insisted Monday that such an agreement was reached, through Egyptian mediation.

And Hamas, for its part, claimed that the drop in the violence stemmed from Israel's realization that its offensive in the Gaza Strip had failed.

Prime Minister Ehud Olmert and Defense Minister Ehud Barak both rejected any suggestion that a cease-fire agreement had been reached with Hamas. They were responding to Monday's Haaretz report stating that the Israel Defense Forces had been ordered to restrain itself in Gaza pursuant to an unofficial agreement with Hamas.

Whatever the reason, the calm continued for a fourth straight day Monday, with no Qassam rockets being fired at Israeli communities from Gaza and no IDF attacks on targets in the Hamas-controlled territory.

Olmert insisted that no negotiations are being conducted with Hamas, either directly or indirectly. Speaking to reporters at the end of a meeting in Jerusalem with visiting Czech President Mirek Toplolanek, he said: "There is no cease-fire agreement; there are no negotiations, direct or indirect."

Barak, who spoke during a visit to the IDF base at Tel Hashomer, said that "at this moment, there is no agreement with Hamas for calm on the ground. When we want to act in Gaza, we shall. When terrorism and arms smuggling into the Strip stop, then we will consider the path toward calm. We are committed to restoring calm to the area and we shall, even if it takes time - and it will involve effort."

However, Barak also spoke of the benefits of calm along the border with Gaza. "If there is school in Ashkelon today without Grad [missiles], and students are in school in Sderot and at Sapir College without Qassams, I suggest that we do not complain about a quiet day."

In contrast to Olmert and Barak, Abbas said Monday that there is a cease-fire agreement between Israel and Hamas.

In an interview with the satellite television station Al-Arabiya, Abbas said that Hamas had asked for immunity from Israeli assassination attempts on its leaders and those of Islamic Jihad, "and I think Israel agreed, or will agree soon."

He said that details of the deal will emerge in the coming days.

Separately, speaking to reporters after talks with King Abdullah II in Jordan, Abbas added that the deal was reached through Egyptian mediation.

"I believe an agreement has been reached in principle" on a cease-fire, Abbas said. "Hamas will stop firing rockets on Israel, and in return, Israel will halt attacks on the Gaza Strip and the West Bank."

He also said that the deal provided for "easing the siege on the Gaza Strip and the reopening of crossing points" into the Hamas-controlled territory.

"Hamas has demanded that its leaders and those of Islamic Jihad be protected against Israeli attacks, and I believe the Israelis have agreed," he added.

Responding to a question about the feasibility of resuming negotiations with Israel at a time when Olmert has approved the building of 750 new housing units in a settlement near Jerusalem - one that Israel wants to keep under any deal - Abbas said: "We consider all settlements as illegal."

"We are going to tell the Israelis that all settlements they build today or tomorrow and all settlements they built after the 1967 war are illegal, and we are going to deal with [them] on this basis," he added.

Abbas said that King Abdullah had expressed support for all international and regional efforts aimed at "helping the Palestinians and Israelis to resume peace talks and find solutions to core issues in accordance with a well-defined timetable."

Meanwhile, Hamas leader Ismail Haniyeh thanked Egypt Monday for its efforts to bring about a cease-fire with Israel, adding that he hoped this would be followed by a deal to mend the rift between the Islamic organization and Abbas's Fatah party.

The Hamas leader also said that the calm stemmed from Israel's realization that its policy in the Strip had failed, and that Hamas had an effective deterrent against Israeli "aggression."