BAGHDAD  After weeks of political stalemate, Iraq approved a law on Sunday to administer a critical national election in January, a significant milestone for its fragile democracy and a step that will allow the rapid withdrawal of American combat forces early next year.

The election, only the second national vote since the fall of Saddam Hussein, will be a crucial step toward popular sovereignty and stability in Iraq. But the election law had been stymied by a political battle over the northern province of Kirkuk, claimed by Kurds, Arabs and Turkmens, each of whom hoped electoral power would give them control of the region’s oil wealth.

The compromise reached Sunday, which satisfied all three groups, was hailed by Iraqi and American leaders as a triumph for Iraq’s emerging democracy and a demonstration of Parliament’s ability to resolve sticky sectarian disputes for the national benefit.

“Accomplishing this law is not a victory for anyone in particular, but a victory for the entire Iraqi people,” said Faryad Raundozi, a member of Parliament’s Kurdish Alliance.