does an increment of 1 degree celsius also causes ice to melt when it is at 0 degree celsius?

an additional amount of heat would be required per unit mass of ice to melt(80kcal/gm).

So here the increase of temperature will help the ice to melt indirectly: the ice would take that extra amount of heat from the environment of higher temp and start melting.It will melt fully or not will depend upon the het supply.