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 $G(p) + βV(x,ω)$, the function $G$ is coercive and strictly quasiconvex, $\min G = 0$, $β>0$, 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 $(θ_1(β),θ_2(β))$, 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 $β$ on $(θ_1(β),θ_2(β))$, and strictly monotone elsewhere.