After 36 years and more than 40,000 deaths, one of the world’s bloodiest and longest-running insurgencies—the separatist struggle of Turkey’s Kurds—could soon be over. Last week Turkey’s Prime Minister Recep Tayyip Erdogan hinted that his government was finally negotiating with Abdullah Öcalan, the leader of the Kurdistan Workers’ Party, or PKK, imprisoned since 1999. Not long ago, such talks would have been political suicide. But Erdogan is riding high after a victory last month in which voters backed his party as it introduced a new constitution pushing the military out of politics. With the Army, traditionally the fiercest opponents of any deals with Kurdish terrorists, on the back foot, Erdogan is now freer to strike a grand bargain with the remains of the PKK.

More important, the militants know they’ve lost the battle for secession from Turkey both politically and militarily. PKK strongholds in north Iraq have been hammered by a series of air and commando raids since 2007. Inside Turkey, too, the organization is losing its grip. A longstanding taboo on speaking out against the PKK was broken last month by the mayor of Diyarbakir, the Kurdish region’s biggest city, who blasted the rebels for a raid on a local stone-cutting factory. Thousands also turned out for the funerals of two popular imams apparently murdered by the PKK, another first in a region where the rebels used to keep the local population in firm check by killing teachers and village elders and forcibly conscripting sons....