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