If any of the glacier is resting on the bottom, this is not true. Obviously a sufficiently large piece of ice will overflow a glass of water when it melts.
I was referring to the melting of the part underwater.
Oh Noes!
Does that mean I'm drinking too fast?
Oh, Yeah...You're wrong.
[Check out how much of that significant ice is below versus how much is above the waterline, then reflect on displacement and on water's expanding when it freezes/contracting when it melts.]