Let $\mathfrak {g}$ be a complex, simple Lie algebra with Cartan
subalgebra $\mathfrak {h}$ and Weyl group $W$. In [MTL], we introduced
a new, $W$-equivariant flat connection on $\mathfrak {h}$ with simple
poles along the root hyperplanes and values in any finite-dimensional
$\mathfrak {g}$-module $V$. It was conjectured in [TL] that its
monodromy is equivalent to the quantum Weyl group action of the
generalised braid group of type $\mathfrak {g}$ on $V$ obtained by
regarding the latter as a module over the quantum group $U\sb
\hbar\mathfrak {g}$. In this paper, we prove this conjecture for
$\mathfrak {g}=\mathfrak {sl}\sb n$.