We present a finite volume method for solving Poisson-Nernst-Planck (PNP) equations in one spatial dimension. To reduce computational cost, an adaptive moving mesh strategy is employed in order to resolve thin Debye layers near the boundary. In addition to the standard monitor functions, we propose two new ones for the moving mesh partial differential equations to improve the accuracy of the numerical solution. Our method guarantees the positivity of the numerical solution and strict mass conservation.