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

The weight of the diamond will cause the average density of the cube to be high enough for it to become fully submerged in the water; displacing the water and making the level rise initially (by the area of the cube). 

Ice itself is less dense than water meaning when it melts the water level would drop slightly as ice takes up roughly 9% more volume than water (when at near 0 centigrade). Although, since the diamond is involved, the cube is not entirely H2O and the diamond remains unchanged. As a result it displaces some water so the water level will actually be the same in the end as it was before the ice melted due to the displacement of the diamond compensating for the differing densities.