Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in
category O. We show that the monodromy of the (normally ordered) rational
Casimir connection on V can be made equivariant with respect to the Weyl group
W of g, and therefore defines an action of the braid group B_W of W on V. We
then prove that this action is uniquely equivalent to the quantum Weyl group
action of B_W on a quantum deformation of V, that is an integrable, category
O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to
V. This extends a result of the second author which is valid for g semisimple.Comment: One reference added. 48 page
