If 50grams of hot water at 80 degree C is poured into a cavity in a
very large block of ice at 0 degree C, what will be the final temp. of
the water in the cavity?  How much ice must melt in order to cool the
hot water down to this temp.?

Since the block of ice is very large temperature can be considered
unchanged overall i.e. 0deg.

The energy carried by the hot water is cooled by the ice and changes
the state of the solid ice to liquid at 0deg. Once this temperature is
reached liquid and solid water will exist in equillibrium

50g = 0.05Kg water @ 80deg needs to cool to 0deg

Using the specific heat of water of 4.186kJ/kgK gives

4.186*80*0.05 = 16.744kJ of energy transfered to the ice.

Using the specific latent heat of ice/water of 334kJ/kg 

gives

16.744/334 = 0.0513Kg of ice that will melt or about 50.1g