No, YOU'RE not listening.
You take a glass, and fill it almost to the top. Then, put a piece of ice into it so that 1/3 of the ice remains on top of the water, and the water level reaches the top of the glass.
Then, when the ice melts, you will see the glass overflow.
Ehh...no, it won't.
Goddamnit, I was hoping I wouldn't have to do this.
Okay, the reason why things float in water, is because they displace water. Whenever water is displaced (or any other substance, for that matter), it generates a force on the object displacing it. Now, if the force is just as large as or greater than the gravitational force on the object, the object stays where it is, or even floats to the surface.
Now, let's look at ice and water, shall we?
The ice displaces an amount of water large enough to make it float. Because of the way ice and water are composed, about two-thirds of the ice will remain under water to provide the necessary force to have the ice remain floating. ("to" is the wrong word, it's not purpose, it's just a simple balance which comes into effect naturally). Now, when that ice melts, it will decrease in volume, but not decrease in mass (this is a unique "ability" of water).
Now, because the ice before only displaced enough water to keep it floating,
the volume of water which was equal to the mass of the ice will have been displaced. When the ice melts, therefore, it will be
the volume of water which was equal to the mass of ice, and that is, in effect,
the same volume which the ice before held in the water. Therefore, the water level
will not rise.
Now, there are two explanations why the water level will rise if the ice melts:
The first is that the ice is not salt, but sweet water, and therefore will have to displace less water (because it doesn't contain salt which would make it "heavier"), and thusly it would indeed take up more "space" if it would melt.
The second is what Big_T said. Thanks.