Archimedes principle states that a floating object displaces it's weight of the fluid that it is floating in, not it's volume. Since the ice cube is approximately 90% as dense as the 0 deg C water that it is floating in, approximately 10% of the ice cube will be above the water surface before it melts. When the ice cube is fully melted, it contributes its weight to the water in the container that it was floating in, which is exactly equal to the weight of the ice cube. This melt water is more dense than the ice cube, such that the "extra" volume of the ice cube that was floating above the liquid surface is no longer there, and that decrease in volume (due to the higher density) exactly matches the "extra" volume that the ice cube contained. Thus, there is no water overflow from the container, due to the changing density and changing volume involved.
If you repeat this experiment by floating an ice cube in salt water, you will see some overflow, so the results of the experiment depend on one or two details of the experiment. For an ice cube floating in salt water of a given salt concentration, that ice cube still displaces its weight of salt water. With salt water being denser than fresh water, the ice cube floats a bit higher in the salt water than it would float in fresh water. However, since the ice cube is composed entirely of fresh water, once that ice cube melts, it adds fresh water to the container, which dilutes the salt concentration in the container. In that case, a more dilute salt water solution takes up more volume than the original salt water, and a small amount of water will overflow the container as a result.