In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson--Nernst--Planck (PNP) system, which has found much use in diverse applications. For the arbitrary high order DG method introduced in [H. Liu and Z. Wang, J. Comput. Phys., 328, 2017], solution positivity is only numerically enforced. In this paper we design a novel DG method, which with Euler forward time discretization is shown to preserve positivity of cell averages at all time steps. Positivity of numerical solutions is then restored by a scaling limiter in reference to positive cell averages. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.