We examine in this paper the evolution of a French discourse marker, a la rigueur, from its original transparent meaning of "rigorously" in late Medieval and Pre-Classical French (15th-16th centuries) to its modern use as a hedge, equivalent to English "at a pinch". We present the results of a large corpus study, using data from various databases covering the diachrony of French, with a qualitative and quantitative analysis of several hundred occurrences. This enables us to show that a la rigueur can be analyzed as having three main use-types, and that its striking semantic evolution results from a very gradual process of semantic evolution via the phenomenon of invited inference (Traugott & Dasher, 2002; Heine, 2002).