Heat Transfer during Phase Change
No, as the ice is melting, the water is cooling down. Ice melts in water because the water molecules in solution collide with those in the ice, giving them energy. In exchange, the water molecules in solution slow down a little during the collision. While the Ice does not change temperature as it is melting, the energy from the water is going into breaking the crystal structure of the ice, turning it into water. This process consumes energy, and the energy to fuel this process comes from the water in solution. Another factor to consider is conservation of energy - if energy is being put into breaking the ice, where else could that energy come from but from the water?