When you put the water in the freezer, it first cools down to its freezing point, then freezes, then keeps cooling to the temperature of the freezer. Once it's been in the freezer long enough to get very close to the freezer temperature, leaving it in longer won't make any difference.
When you take the ice out into a warm environment, the whole process reverses. If the freezer was colder, then the ice has a little more warming to do before it reaches the melting point, so it will take longer before it melts. However, the effect is not as large as you might expect. It doesn't take much heat to warm the ice up to the melting point, compared to the heat required to actually convert the ice to liquid while it's melting. Specifically, it takes 334 Joules of heat to melt one gram of ice already at the melting temperature (0°C). For every degree Celsius that the ice starts below the melting temperature, you need only about another 2 J. With a household freezer, that won't amount to much extra, since I doubt you can set it below -20 °C, if that low.
(republished on 07/25/06)