I'm a little confused. (nothing new there then)
The article seems to be discussing a sheet of 'marine ice' which has broken off, and is expected to melt.
Based on its size they give a prediction of a rise in sea levels.
I thought (and it seems Archimedes agreed) that "A body wholly or partly immersed in a fluid receives an upthrust equal to the weight of the fluid displaced", so ice floats only because it is less dense than the water below it.
If it melts, then in the process it becomes more dense, and contracts to replace the volume of water it previously displaced. As such, apart from any slight difference in densities of fresh/salt water, why would this cause an increase in sea levels?

Try this at home, put a few ice cubes in a glass, and fill to the brim with water. When the ice cubes melt does the water suddenly overflow the glass?