Shortly after 9 a.m. on June 4, three men drove to a seaside promenade near Marseilles, their van carrying paintings by Brueghel, Sisley and Monet. The art had been stolen at gunpoint from the Museum of Fine Arts in Nice last August. Now a Frenchman working for an American art dealer was supposed to show up and buy four works for $4.6 million in cash. Instead, nearly a dozen French police cars pulled up, led by a colonel for the gendarmerie who quickly took a call from Pennsylvania. "We got them!" Col. Pierre Tabel shouted into his cellphone.

The...