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The dispersion properties of the guided modes propagating along a dielectric slab waveguide with a left-handed material substrate are studied. It is shown that both the oscillating and surface modes can propagate very slowly along such a waveguide if the thickness of the core layer is chosen appropriately. The propagation speed of the guided waves can even approach zero and this slow propagation is illustrated with an finite-difference time-domain simulation in a tapered waveguide. The influence of the material loss on the slow propagation is also discussed.