I think it would go down... here's why.

When water freezes, it expands. That's what full bottles in a freezer burst, as do pipes in winter. So 1 gram of water occupies less space in the glass (1 ml) than 1 gram of ice.

According to Archimedes' Principle, a floating object will displace precisely as much water as its weight. Which is why a log of wood floats - the water it displaces weighs more than the wood itself.

If 1 gram of ice is floating, it means it has already displaced 1 gram of water. When 1 gram of ice becomes 1 gram of water, it will occupy less space.

As the ice melts, the level of water comes down.