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
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...
- Water actually contracts when its temperature rises from 0°C
to approximately 3.98°C.
(Beyond that point, water expands when temperature rises, just like most substances do.)
- The cross-section of the container increases with temperature, so that a volume
of water which decreases or remains constant will have less height.
This argument shows that the water level will continue to decrease (at least initially)
at the 3.98°C point.