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The density of the water will remain constant whilst the water remains in one state.
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This is an incorrect statement - 'fundamentally flawed', in fact.
http://www.simetric.co.uk/si_water.htm
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The density of pure water is a constant at a particular temperature, and does not depend on the size of the sample. That is, it is an intensive property. The density of water varies with temperature and impurities.
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It's perfectly obvious that melting sea ice will not cause a rise in sea levels, just as it's perfectly obvious that melting land ice *will* cause a rise in sea levels.