Can You Solve this Ice Diamond Riddle? ft. Vsauce’s Michael Stevens

Taking everything into account: It depends, we'll have to know all the specifics to answer this. 

This is why:
Parts of the ice cube are slightly above the surface, so it has a lower density than the water. If the top bit, while dissolving, adds more water than the bottom bit loses volume when it dissolves, then level in the end will be higher. The diamond will not melt and add a fixed amount from the top and keep a fixed bit from the bottom (though the diamond is probably wholly under the surface, as that is the stable position) .
If there is no icy bit at the top and ice/water have the same density, then nothing changes. 
If the cooling ice reduces the water temperature towards 4°C, where it has minimum density, and the top bit adds less volume than the bottom bit loses, the level decreases. There are some more cases, but I think it boils down to stuff being inconclusive (pun unintended) :')