When Ice floating in water dissolves, what will be impact on the level of water when the ice melts away completely?

There are two dimensions of this experiment. One is when Ice floats in water and another is as mentioned in the question. When Ice floats in water, the level of water will remain unchanged when the ice melts away because in this situation, Ice replaced water which is ""equal to its weight"". But when the floating ice is taken down in the bottom of the bucket and tied over there with a string, and let melt away, the level of water will come down because now the ice was replacing the water ""equal to its volume "" and Ice has a lower density of water.