Astract We study here the ring QSn of Quasi-Symmetric Functions in the variables x1, x2, . . . , xn. F. Bergeron and C. Reutenauer [4] formulated a number of conjectures about this ring, in particular they conjectured that it is free over the ring Λn of symmetric functions in x1, x2, . . . , xn. We present here an algorithm that recursively constructs a Λn-module basis for QSn thereby proving o...