Absolutely nothing would happen, as far as you would be able to tell. The ice and the water would dwell in peaceful coexistence. But down at the molecular level, a chaotic dance would be going on.
Zero degrees Celsius (32 degrees Fahrenheit) is indeed both the freezing point of liquid water and the melting point of solid ice. You are undoubtedly picturing a poor little zero-degree water molecule that can’t make up its mind whether to flow or float, to be liquid or solid. Well, that’s really a good way to look at it, because the individual molecules do indeed get to make decisions, in a manner of speaking.
Let’s consider first what goes on when liquid molecules freeze. There are some rather strong attractions between water molecules that tend to make them stick together. (In Techspeak, they are called hydrogen bonds and dipole-dipole attractions.)
In liquid water, the molecules are moving fast enough that the attractions can’t really take hold. But as water, or anything else for that matter, cools, its molecules move more and more slowly. Zero degrees Celsius happens to be the temperature at which the molecules are moving just slowly enough that they can grab hold of one another with their attractive forces and settle down into the unique fixed positions that characterize ice. Ice’s molecules are tied rigidly in place; they can’t go swimming around the way liquid molecules do.
Now let’s put an ice cube into liquid water. Some of the ice molecules at the surface of the ice will break their attachments to their fellows and join their freely swimming brethren. In other words, they will melt. Meanwhile, some of the liquid molecules near the ice’s surface may be moving more slowly than the capture speed (they’re not all moving at the same speed), and they will freeze onto the ice. So both melting and freezing can be taking place simultaneously, some molecules going one way and some going the other.
As long as the water is slightly warmer than zero degrees Celsius, there will be more melting going on than freezing, because there won’t be enough slow water molecules to be captured onto the ice. Conversely, if the water’s temperature is slightly lower than zero degrees, there will be more freezing going on than melting, because there will be more slow water molecules to be captured.
At exactly zero degrees, there will be just as many ice molecules melting as there are liquid molecules freezing. Millions of tiny molecules are going each way, but from our relatively gigantic human perspective, we see absolutely nothing going on. The ice and water just sit there, until, of course, they begin to warm up, and then melting takes over.
In a sense, then, zero degrees Celsius is neither the “melting point” nor the “freezing point” of water. It’s the temperature at which melting and freezing are happening equally. Scientists call this exact-balance point an equilibrium point. They would say that at zero degrees Celsius, ice and liquid water are “in equilibrium.”
Equilibrium is a very important concept in chemistry because there are many situations in which, down at the molecular level, two opposing processes are going on at equal rates, so that up here at the human level we see no apparent change.
For more, look up “equilibrium” in the index of any chemistry book. But I warn you: There may be lots of equations.
Melting and freezing are so closely interrelated that just by touching an ice cube you can kick some water molecules from liquid to solid.
Wet your fingers and touch the ice cubes in your freezer. The cubes may stick to your fingers so tightly that you can lift them up. The ice cooled the water on your fingers down to its own temperature, which is obviously below the freezing point. When the water on your fingers froze, it grabbed on to the ridges of your fingerprints and held on to them while simultaneously fusing itself onto the ice cubes, thereby “gluing” your fingers to the cubes.