Baladi V., Ruelle D. An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps (45K, AmS TeX Version 2.1 (Postscript figure)) ABSTRACT. We consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor-Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.