In May of last year, a small team of international weapons inspectors gained entry to one of Syria’s most closely guarded laboratories. Western nations had long suspected that the Damascus facility was being used to develop chemical weapons.

Inside, Syrian scientists showed them rooms with test tubes, Bunsen burners and desktop computers, according to inspectors. The Syrians gave a PowerPoint presentation detailing the medical and agricultural research they said went on there. A Syrian general insisted that the Assad regime...