In this paper, we study a double barrier option when the underlying asset price follows a regime-switching exponential mean-reverting process. Our method is a combination of analysis of a deterministic boundary value problem with a probabilistic approach. In this setting, the double barrier option prices satisfy a system of m linear second-order differential equations with variable coefficients and with Dirichlet boundary conditions, where m is the number of regimes considered for the economy. We prove the existence of a smooth solution of the boundary value system by the method of upper and lower solutions; we proceed to construct monotonic sequences of upper and lower solutions that converge to true solutions, respectively. The uniqueness of the solution is established by applying Dynkin's formula. This proof by construction also provides a numerical procedure to compute approximate option values. An important feature of the proposed numerical method is that the true option values are bracketed by the upper and the lower solutions. Examples are provided to illustrate the method.