Normally, the water stays near 0 deg C, because if got any colder the ice would start to grow. The ice has lower potential energy than the water, so as it forms energy is relesed, warming the water. So it stays at near 0, even if heat flows out to the surroundings, until all the water is frozen.
If you put salt in ice-water, it lowers the freezing/melting point. We’ve discussed that in other answers. So if it starts near 0 deg C, the ice is above the new melting point, and starts to melt. As it melts, it soaks up energy- for exactly the same reason that it dumps out energy when it freezes. The amount of energy needed to melt ice is 80 calories per gram. That energy comes from the water, cooling it. Removing one calorie from a gram of water reduces its temperature by one degree C. That cooling goes on until the ice is all melted or until the temperature drops to around the new, lower melting point- say - 8 deg C, or even lower if the water is salty enough.
(published on 10/22/2007)