The confrontation between the Palestinian leader and the premier, Ahmed Qureia, centers on whether Arafat is willing to cede any of his absolute authority. Israel considers Arafat's stranglehold on Palestinian affairs as an obstacle to Mideast peace.
At the end of an emergency meeting in Arafat's ruined West Bank headquarters, the two men deadlocked over who would have ultimate control over the security services and whether Qureia would stay in his job.
"President Arafat insisted in rejecting the resignation. Abu Ala insists on his resignation. The crisis goes on," Saeb Erekat, a senior Cabinet minister, told The Associated Press, using Qureia's common name.
Other ministers said Qureia ended the meeting by telling Arafat, "my resignation stands. I consider my government a caretaker government."
The Cabinet crisis exploded after a series of kidnappings and violent demonstrations in Gaza last week, followed by Arafat's reshuffle of top security posts. He appointed his widely disliked cousin, Moussa Arafat, to the top security job in Gaza.
The prime minister submitted his resignation on Saturday, protesting the upheaval in Gaza and the disorder in the Palestinian security services. On Monday, Arafat reinstated the officer his relative replaced — Abdel Razek al-Majaide — but retained Moussa Arafat in a powerful position, satisfying some of his critics but infuriating others.
The armed chaos in the Gaza Strip is posing the biggest threat to Arafat's rule since he took over the Palestinian territories 10 years ago.
"This cannot be tolerated under any circumstances, and I believe restoring public order and the rule of law is the priority," said Palestinian Cabinet Minister Saeb Erekat.
CBS News Correspondent Robert Berger reports Gaza is ruled by rival militias, and the Palestinian Authority fears the worst case scenario: a Palestinian civil war.
Meanwhile, Israeli Prime Minister Ariel Sharon says the turmoil in the Gaza Strip is proof that Israel has no one to talk peace with on the Palestinian side.
"There is nobody to negotiate with, there is no partner," Sharon said.
Sharon sees that as giving no choice but to implement his plan to pull out of Gaza unilaterally, but it's that very plan that's sparked the current power struggle in Gaza. The Palestinian Authority and militant groups are vying for control.
Sharon, too, is having his own problems over the pullout plan. He lost his parliamentary majority because of opposition in his own government to the proposal. Now he is approaching opposition parties to join his coalition, to restore stability.
In a second day of talks with Sharon's team, Labor Party representatives said Monday they want the parliament to quickly approve compensation for settlers who are evacuated — a sign that Sharon is serious about implementing the plan.
Labor favors giving up all or most of the West Bank and Gaza in exchange for peace with the Palestinians — a position opposed by many members of Sharon's own Likud Party.
Likud representatives met with the ultra-Orthodox rightwing Shas party Tuesday in an effort to bring it into the ruling coalition. It may just be a negotiating ploy with Labor, however, because Shas has reservations about the Gaza withdrawal plan.
The head of Israel's internal security service, Shin Bet, told parliament Tuesday that about 150-200 Israeli Jews are "actively awaiting" the death of Sharon. Avi Dichter did not clarify what that meant.
Dichter said that if the people on the list, most of them settlers in the West Bank and Gaza, were Palestinians, they would be placed in administrative detention.
Dichter said earlier this month that "the activities of Israel's extreme right have escalated."
On Monday, a gunman shot and killed a district court judge in a Tel Aviv suburb, police said. Israel's justice minister said it was the first such killing in Israel's history.
The 49-year-old judge, Adi Azar, was in his car near his home in Ramat Hasharon when he was shot from close range by a man on a motorcycle, witnesses told police. The assailant, described by police sources as a "mafia-style professional killer," escaped.
The motive for the killing was unclear. The Al Aqsa Martyrs' Brigades in the West Bank claimed responsibility, but Lapid said there was nothing to support the claim.