  The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of
projective, genus $6$, complex-algebraic curves admitting a faithful action of
the icosahedral group $\Af_5$. The goal of this paper is to prove that the
monodromy of $\Cs/\mathcal B$ is commensurable with a Hilbert modular group; in
particular is arithmetic. We then give a modular interpretation of this, as
well as a uniformization of $\mathcal B$.
