We propose the characteristics-mixed method for approximating the solution to a convection-dominated transport problem. The new method is a combination of characteristic approximation to handle the convection part in time and a mixed finite element spatial approximation to deal with the diffusion part. Boundary conditions are incorporated in a natural fashion. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Analysis shows that the scheme has much smaller time-truncation errors than those of standard methods.