[iamkroot]

The ice actually comes up to the temperature of the water while it's melting. That's what the equilibrium temperature is: the temperature of the entire ice / water system until it's been converted to all liquid or all solid.

Naturally there's some small local variations, but if you let the system come up to steady state, that's what will occur.

[topaz0]

To be fair to GP, it does take some energy to heat the ice from freezer temp to 0C (or the new, depressed freezing point), part of which will come out of the ice cream. It's just that that amount of energy is very small compared to the other energies we're interested in here (as you pointed out elsewhere).

[Taniwha]

That's true of the surface of the ice, but the core is colder

[iamkroot]

For a little while, yes. Ice has middling thermal conductivity, it'll eventually homogenously warm to the melting point.