Israel's military campaign in the Gaza Strip is failing, but there may be a silver lining. The war against Hamas is proving — once again — that the Middle East's extremist movements cannot be eliminated by military means. If the incoming Obama administration absorbs that lesson, it will have a better chance of neutralizing Iranian-backed groups such as Hamas and Hezbollah, and of eventually brokering an Israeli-Palestinian peace settlement.

Israel's bet was that it could substantially reduce Hamas' military capacity and then force it to accept a cease-fire with improved terms for Israel. Hamas, predictably, has refused to play by those rules. It has defined victory as its own survival; by that standard, it has no incentive to agree to a new truce unless it receives major benefits in return, such as an end to Israel's economic blockade.

That means Israel must choose among attempting to drive the Islamic movement from power (which would be costly and leave its troops stuck in Gaza indefinitely), making concessions to Hamas or withdrawing without any assurance that rocket fire against its cities would cease.

At best, outgoing Prime Minister Ehud Olmert might win an agreement for international forces to help stop the smuggling of new weapons from Egypt into Gaza, something that doesn't necessarily require Hamas' consent. But that won't stop Hamas from continuing to build its own rockets or from claiming that — like Hezbollah in Lebanon — it successfully resisted an Israeli invasion.

The trap that Olmert, Defense Minister Ehud Barak and Foreign Minister Tzipi Livni have created for themselves lies not just in Hamas' ability to withdraw its fighters and rockets into mosques, schools and densely populated neighborhoods, where they could probably survive weeks of bloody fighting or go underground. The larger fallacy is the persistent conceit among Israeli leaders that Hamas can somehow be wiped out by economic strangulation or force of arms.

Unlike al-Qaida, Hamas is not merely a terrorist organization but a social and political movement with considerable support. Its ideology, however repugnant to Israel and the West, is shared by a considerable slice of the population in every Arab country from Morocco to Iraq. Because it is extremist, it thrives on war, the suffering it inflicts on Palestinians, and the anger generated by the endless, graphic and one-sided coverage of the Middle East's satellite television channels. Every day this war continues, Hamas grows politically stronger, as do its allies in other countries and its sponsor, Iran.

Though Israel must defend its citizens against rockets and suicide bombings, the only means of defeating Hamas are political. Palestinians, who have no history of attraction to religious fundamentalism, have to be persuaded to choose more moderate leaders, such as the secular Fatah. In the meantime, Hamas' existence must be tolerated, and it should be encouraged to channel its ambitions into politics rather than military activity. That means, yes, elections — like those Hamas won in 2006, when it took control of the Palestinian legislature.

Those elections took place over Israel's objections, and the outcome caused the Bush administration, which had championed democracy in the Middle East, to lose its nerve. But during the relative quiet of the past six months, when Israel and Hamas observed a semi-truce, politics was beginning to work. Polls conducted by Palestinians showed that Hamas' support was falling in Gaza and the West Bank. Mahmoud Abbas, the Palestinian Authority president and Fatah leader, was beginning to talk about holding new elections for president and the legislature; he thought he could win both.

A split began to emerge in Hamas between leaders who wanted to make that deal and extend the peace with Israel, and Iranian-backed hard-liners who wanted to draw Israel into a fight. Israel probably could have ensured that the moderates won by offering to lift its economic blockade of Gaza in exchange for a continued cease-fire.

Israel took the Iranian bait and chose to fight. Now, bogged down, suffering casualties and inflicting many more, creating terrible pictures for television, it will have to accept an unsatisfying settlement — or prolong its agony indefinitely. It should settle so that the leaders chosen by Israeli voters in an election next month will have the chance to work with a fresh American administration on a smarter and more effective strategy for countering Iran and its clients — one grounded in politics rather than bombs.

Jackson Diehl is deputy editorial page editor for The Washington Post.