  We prove homogenization for a class of viscous Hamilton-Jacobi equations in
the stationary and ergodic setting in one space dimension. Our assumptions
include most notably the following: the Hamiltonian is of the form $G(p) +
\beta V(x,\omega)$, the function $G$ is coercive and strictly quasiconvex,
$\min G = 0$, $\beta>0$, the random potential $V$ takes values in $[0,1]$ with
full support and it satisfies a hill condition that involves the diffusion
coefficient. Our approach is based on showing that, for every direction outside
of a bounded interval $(\theta_1(\beta),\theta_2(\beta))$, there is a unique
sublinear corrector with certain properties. We obtain a formula for the
effective Hamiltonian and deduce that it is coercive, identically equal to
$\beta$ on $(\theta_1(\beta),\theta_2(\beta))$, and strictly monotone
elsewhere.
