Because of the presence of the ice cube, the water is at 0°C. Since the icecube is floating, the part of it that's below the surface of the water has a volume equivalent to the volume occupied at 0°C by a mass of water corresponding exactly to the mass of the ice cube (Archimedes' Principle). As the cube melts and turns into water at 0°C, it will therefore occupy exactly its former "underwater" volume (nothing more, nothing less) so that the level of the water will remain the same!

When the ice cube is completely melted, however, the temperature of the water may rise above 0°C. Shall we then expect the water level to rise or fall? Actually, it will then fall for two reasons:

Of course, we assume that the level is measured from the bottom of the container with instruments not sensitive to temperature (beware of rulers). The contraction is minute and will not be observed in the kitchen, but this would be readily measured in the lab with precise enough instruments... When the cube was not yet fully melted, the temperature remained constant (0°C) and even a temperature-sensitive ruler would have "shown" the level to remain constant...