I actually JUST did this quest on my mage yesterday. It starts with the quest that you ignite the bomb that sends you under Icecrown Citadel and find Arthas's Frozen Heart.

The final part of that chain (Tirion's Gambit) ends with Tirion destroying the heart after he officially decides that Arthas has no humanity left.

I'm inclined to disagree though, at the end of the Fall of the Lich King video, I think that with Frostmourne broken and the Helm of Domination removed, that he became a bit more human. He seemed happy that it was over, in my opinion.

Or maybe it's just because he was dying.