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

The water level will clearly fall. It is easier to understand this by looking at the extreme case. Imagine we have a blackhole encased in the ice instead. The same volume of ice cube will displace a lot of water resulting in the water level being very high (assuming the ice cube floats because remember the actual question is with a piece of diamond and the ice is actually floating in that situation.

However, after the ice melts the black hole will sink and essentially take up no volume (being a blackhole and all) while the volume of water that is added back from the ice is just the volume of the ice which is much less than what was originally displaced. So the water level will clearly fall.

Not convinced? we can see it in another way as well. Lets break up the problem. What if we have ice with no diamond inside. The ice has a mass of m and displace water of mass m to stay a float, when ice melts the ice melts into water of mass m and hence no change in volume. so now if we introduce a diamond in the ice, the ice essentially act as a boat in the boat problem in another of Diana's video. 
Reference here: https://www.youtube.com/watch?v=nVT1c0tr8NE