In the two years since it seized power here, the militant Hamas movement has undercut the influence of the Gaza Strip's major clans, brought competing paramilitary groups under its control, put down an uprising by a rival Islamist group, weathered a three-week war with Israel, worked around a strict economic embargo -- and through it all refused a set of international demands that could begin Gaza's rehabilitation.

That combination of durability and unwillingness to compromise has created a deep-seated stalemate that has left top Israeli intelligence and political officials perplexed about what to do, and it has posed a steep obstacle for U.S. peace envoy George J. Mitchell. Mitchell's work in Northern Ireland in the 1990s included intense negotiations to bring the most militant parties into the process, but his eight months of talks about Israeli-Palestinian peace have avoided any obvious effort to do the same with Hamas and have been conducted, in effect, with only one half of the Palestinian political leadership.

A separate Egyptian effort aims to reconcile Hamas and the pro-U.S., West Bank-based government of Palestinian Authority President Mahmoud Abbas, create a joint security force and pave the way for elections next year that could bring Palestinian society under a single political leadership. But Palestinian, Israeli and international diplomats and analysts give the process only a slim chance of success and see little sign that Hamas is ready to trade its clear control of the Gaza Strip for a seat at the negotiating table.

Barack Obama's election as U.S. president and his June speech in Cairo raised expectations among Hamas officials of a dialogue with the United States, but "people are starting to lose hope. There was a glimmer, but it is fading away," said Hamas deputy foreign minister Ahmed Yousef, adding that Mitchell's work has produced "no solution and no breakthrough."

A top Israeli security official said there has been a frustrated acknowledgment in Israeli intelligence and military circles that, as it stands, there is no obvious alternative to continued Hamas rule in Gaza. The Palestinian Authority is not strong enough to return to power there, Israel does not want to reoccupy an area it vacated in 2005, and there is concern that any collapse of Hamas rule might increase the influence of even more militant groups.

"We don't like them, but they have accountability," said the official, adding that Hamas is struggling to reconcile running a government and staying in power without losing its credentials as a resistance movement. At present, he said, the group is trying to maintain a policy of what the Israeli military refers to as "industrial quiet" -- suppressing most rocket fire into Israel as part of a pause in violence that is practical, for rearming, and strategic, to ensure its hold on power. How, when and whether Hamas might tip back toward fighting is uncertain. When diplomats, outside negotiators and others ask for ideas about how to cope with Hamas in the long term, the Israeli official said, the answer is: "We don't know. Good luck."

Hamas, which was founded as an Islamist alternative to the Palestine Liberation Organization and whose charter calls for Israel's destruction, is considered a terrorist group by the United States for its sponsorship of suicide attacks and the launching of thousands of missiles and mortar shells from Gaza into Israel. The group draws financial and material support from Iran and Syria. Hamas says its attacks on Israel are defensive and a legitimate tactic in Palestinian efforts to establish a homeland.

Mitchell faced a similar dilemma during the Northern Ireland peace process, when there was opposition to the inclusion of the Irish Republican Army in the talks and demands that the group disarm before becoming party to the discussions.

But as Mitchell and fellow peace envoy Richard N. Haass would later write, "It's hard to stop a war if you don't talk with those who are involved in it."

In the case of Northern Ireland, Mitchell argued as a senator for IRA leader Gerry Adams to be granted a U.S. visa. As a peace envoy, Mitchell took a controversial stand in favor of letting the IRA retain its weapons while joining the peace talks -- evidence of his belief that "preconditions ought to be kept to a minimum."

But the group had to endorse what came to be known as the "Mitchell principles" of democracy and nonviolence. Similarly, there has been a standing offer from the United States and other nations to reopen talks with Hamas if the group meets certain conditions, including a renunciation of violence, adherence to prior agreements made on behalf of the Palestinians and a recognition of Israel.

According to officials from Hamas and analysts of the group, those conditions are unlikely to be accepted, cutting as they do to the core of the group's ideology and strategy. Just as there is no sense that the language of Hamas leaders has come close to meeting those requirements, despite talk of a possible compromise, there has been no obvious effort by Mitchell's team to try to reshape the conditions.
"Nobody has really grabbed hold of the issue of what we do with Hamas," one Western diplomat said.

Members of Mitchell's team would not comment on the issue.

The assumption of the government of Israeli Prime Minister Binyamin Netanyahu is that deteriorated conditions in Gaza will undercut Hamas's popularity, particularly as people learn of improvements in the lives of Palestinians in the West Bank. Gazans are isolated in a subsistence economy with large-scale unemployment. Membership in Hamas is considered a prerequisite for public jobs. The group's armed wing, the Izzedine al-Qassam Brigades, proved ineffectual in last winter's Gaza war, unable to either protect the population or inflict enough Israeli casualties to score a propaganda victory. Disenchanted residents speak of "cutting off their thumb" -- a reference to the ink applied on voters' fingers to indicate they had cast ballots in the 2006 elections, in which Hamas trounced Abbas's Fatah party.

Still, there is little expectation that any sort of popular uprising will challenge Hamas, and recent events hint at why the group is biding its time.

"Yes, people are not satisfied because of the division. The economic situation is bad. They are looking for change," said Yousef, the Hamas official. But "they also understand why we are suffering."

In the past two weeks, Mitchell has scaled back U.S. demands for Israel to freeze West Bank settlements and persuaded a reluctant Abbas to attend a meeting with Netanyahu in New York.

A flare-up of violence around the al-Aqsa mosque in Jerusalem has been trumpeted by Hamas as proof that the peace process won't protect Palestinian claims on the holy city. The Palestinian Authority's decision to back off of demands for investigating allegations of war crimes in the recent Gaza conflict, meanwhile, has created a deepening crisis for Abbas as he fends off criticism throughout the Arab world that he is protecting Israel without winning any political concessions.

Just as the "peace culture" that followed the signing of the Oslo accords in the early 1990s gave way to an intifada, or uprising, in 2000, Hamas officials say the current upbeat mood in the West Bank will at some point erode -- leaving groups that support armed action again in favor.

Fatah "considers peace a strategic choice. That is why they are always working on it," Hamas spokesman Ayman Taha said. After Oslo, "there was seven years of peace culture, and at the end everything turned upside down. And what is going on in the West Bank now is temporary."

It has been long-standing Hamas policy to consider a long-term ceasefire with Israel in return for establishment of a Palestinian state on the Gaza and West Bank land occupied by Israel in the 1967 Arab-Israeli war.

But that "is only a way to kick the occupation out," Taha said. "It is a staged or phased solution, which is the 1967 borders, and a strategic objective to bring back all the territory occupied in 1948," when the state of Israel was created.

"If the international community agrees to a full state on the borders of '67, then we will decide what to say at that point," he said. "It is still early."