Born in Le Puy-en-Velay, Haute-Loire, Coupet has 34 caps with France. He made his international debut against Australia during France's 2001 FIFA Confederations Cup-winning campaign. He was called up for the 2002 World Cup as the backup to Fabien Barthez, but did not play. The following year, Coupet kept three clean sheets and conceded only three goals in five games as France won the 2003 Confederations Cup, which marked their second victory in the competition in three years. During a commemoration for the late Cameroon player Marc-Vivien Foé, Coupet broke down on tears, since they both were teammates while playing for Lyon in the French league.