We give herein analytical formulas for the solution of the Hermite collocation discretization of the unforced steady-state convection-diffusion equation in one spatial dimension and with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. The accuracy of the method is advanced by employing "upstream weighting" of the convective term in an optimal way, avoiding both "smearing" and unwanted oscillations, particularly for large Péclet numbers. Computational examples illustrate the efficacy of using optimal upstream weighting.