Compare the mass of the ice cube with the mass of water vapor

Hmmm Yes ice does have the same mass as water before it melts. You can (almost) never lose or gain mass without adding or subracting mass from the system. Mass just can't appear or dissappear from nowehere. 
It is known as the law of conservation of mass. 

In water though, the volume that ice occupies is greater than that of water due to the arrangements of the molecules of ice. This is why you would be led to think ther is a change in mass. 

A good weigh to prove this to yourself would be to get an ice cube in a bowl and place it on a set of scales as it melts. You shouldn't notice any real mass change. 

p.s. my almost never is because that nuclear reactions can turn mass into energy through Einsteins equation E=mc2 but that is rare,