The Muskingum-Cunge scheme applied to the one-dimensional unsteady advection-diffusion equation is investigated. To eliminate the numerical diffusion, the coefficients of the scheme are defined in such a way that the scheme does not contain the weighting parameters explicitly, but the Courant and Péclet numbers only. If one of the weighting factors is prescribed, the other should be necessarily negative in a lot of cases, which does not affect the applicability of the scheme. It is shown that the accuracy can be increased further, the numerical oscillations can also be eliminated by prescribing a simple relationship between the Courant and Péclet numbers. Sufficient conditions for strong stability are also presented.