Iran and America must also manage external spoilers. Saudi Arabia has long seemed unwilling to accept the realities of the new Iraq. But the kingdom can be flexible when intransigence seems self-defeating. Washington must impress upon the Saudis that the fire of extremism will inevitably enter the heart of the Arabian Peninsula unless action is taken to halt support for militancy. Indeed, in a twist of calculations, America may actually now share Iran’s interest in seeing ISIS’s other major foe, the Syrian regime of Bashar al-Assad, go after Sunni extremists. Mr. Assad’s warplanes are now bombing militants on the Iraqi border, which they were not doing last week.

Iran and the United States should also seek to divide ISIS. If the group is only confronted in Iraq, it will withdraw to Syria to return another day. The United States can’t and shouldn’t act as Iraq’s air force. But American military and technological prowess — in the form of sales of drones, helicopters and fighter jets — should be combined with Iranian and Syrian intelligence to prevent the movement of extremists.

Finally, Iran and the United States must boost the Iraqi Army’s strength and prevent the rise of militias. Mr. Maliki claims that thousands of volunteers who have signed up to fight ISIS will be the core of the next Iraqi Army, but he needs enough political, military and intelligence help from America and Iran so that he won’t have to rely on irregular forces. Any shift away from the army and toward the militias would be profound and unpredictable.