[iamkroot]

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

[Taniwha]

yes - then it wont be ice any more - but while there's still ice the temp of the liquid will stay at the equilibrium point - after that it will start to warm

[topaz0]

"then it won't be ice anymore". Not so: the freezing point is the temperature at which ice and water are in equilibrium. If you take a bowl of ice to its freezing point it will still be entirely frozen. Then if you continue adding energy, it will remain at its freezing point while progressively more of it melts, i.e. going from 100% ice to 50% ice/50% water to 100% water. All at one temperature. In other words, if you are at the freezing point, then all of the energy you add goes to melting the ice (and none of it goes to increasing the temperature), until all of the ice is gone. That is the case even if you wait to true equilibrium, e.g. for all temperature gradients to go away.