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If you wanted to kill someone with a bullet made of ice, or, IMHO even cooler, dry ice, you'd forgo a conventional cartridge, which would contaminate the scene with gunpowder anyway.

You'd use something like compressed air. You can reduce the instantaneous pressure delivered to the projectile which helps fix the melting problem. You'd want to chill the barrel a bit, but some melting might be good there as long as the water doesn't refreeze in the barrel and stop successive rounds. Basically you're making a potato cannon that shoots ice.

Bonus for freezing the water/co2 out of the air itself. You wouldn't even need to carry ammo.