Shortly after the end of World War I, the French and British prime ministers took a break from the hard business of redrawing the map of Europe to discuss the easier matter of where frontiers would run in the newly conquered Middle East.

Two years earlier, in 1916, the two allies had agreed on their respective zones of influence in a secret pact—known as the Sykes-Picot agreement—for divvying up the region. But now the Ottoman Empire lay defeated, and the United Kingdom, having done most of the fighting against the Turks,...