We consider the exit-time problem for a particle diffusing in a one-dimensional random medium with a global nonrandom bias field. It is found that, for small bias, in addition to the usual renormalization of the diffusion coefficient the fluctuations in the hopping rates have the effect of increasing the mean exit-time, independently of the direction of the bias and the initial position of the particle. We also introduce an effective-medium approximation that becomes exact in the limit of small bias.