We have calculated dispersion coefficients for tracer particles in a fluid flowing through a porous medium consisting of randomly packed spheres. At high Péclet numbers, where the motion of the tracer is determined largely by convection, we found evidence that the dispersion coefficient is diverging and that the dispersion process is anomalous. A transient region of anomalous dispersion has been predicted theoretically. However, our simulations suggest that, rather than being transient, this effect persists. We argue that our findings are consistent with the available experimental data.