From Strasbourg, in Alsace, working in Paris, Weyler was considered the finest miniaturist of his day. His name is also spelled Weiler. He worked in the studio of Joseph Marie Vien from 1763, and was admitted as an associate of the Royal Academy in 1775. He exhibited regularly at the Paris Salon, and is cited as having painted portraits of many prominent figures. At least one existing enamel portrait of Franklin is attributed to him.
In 1785 he was commissioned to create a "panthéon iconographique" of famous men, both French and foreign. Franklin was included and was very popular, as Weyler created additional versions, for a total of five in enamel and two on ivory. One in enamel is on display in the Gilbert collection at Somerset House, London.