A two-layer dielectric system, consisting of a thin polystyrene sheet on top of water, is examined as a possible guiding structure for surface waves. Experimental investigations, at a frequency of 9.35 Gc/sec, of the phase velocity of the waves close to the surface of the upper layer, and the power variation with height above it, are described. A slow wave was found to propagate near the surface and its phase velocity was found to agree with that predicted theoretically for a TM surface wave. Qualitative agreement between the experimental and theoretical power variation with height confirmed the existence of a surface-wave contribution to the total field above such a layered dielectric system.