At the beginning of the 17th century. This point in time is fairly accurate, but who invented it, is rather unknown.
A legend exists though, famous in the region, of the Chevalier de la Croix-Marron (the knight of the Brown Cross) who allegedly invented double distillation on account of a dream. Chevalier Jacques de la Croix-Marron really existed. He was a hugenot who fought together with Agrippa d’Aubigné during the religious wars; he also wrote the book La Muse catholique, a two-piece book: Libre Arbitre and L’Eucharistie
According to the legend the knight dreamt of the devil who boiled him in his cauldron to extract his soul. But the knight’s soul resisted strongly and the devil had to cook him again. When the knight woke up, he remembered his dream and got the idea of extracting the soul of eau-de-vie by distilling twice.