This is the same issue documented in this thread in Insufficient Information:

http://forum.square-enix.com/ffxiv/t...losing-on-Exit

and this thread in Technical Support:

http://forum.square-enix.com/ffxiv/t...y-since-update

Time: Any, since the latest hotfix patch

Frequency: Sometimes

World name: Ultros

Character name: Orexis Guile

NPC name: none

Monster name: none

Class/Level: Dragoon/50

Party or solo: solo

In-game time: don't remember

Area and coordinates: In any sanctuary or inn

Connection Specs
- Type of internet connection/provider: Cable/Charter

PC Specs
-=-=-=-=- System Information -=-=-=-=-
Processor: AMD Phenom II X4 965 3.40 Ghz
RAM: 8 GB
OS: Windows 7 Home 64-bit
Video Card: EVGA Nvidia Geforce GTX660 2GB

Steps:
1. Load the game and play for ~2 hrs
2. Exit the game in a sanctuary or using the bed in any inn
3. Screen goes black until only the mouse cursor remains (cursor is movable), however, the game does not close
4. I have to alt-tab out and then ctrl-alt-del to access Task Manager and then close the program via Task Manager

Other notes:
This has happened to me twice since the hotfix Monday. Both times I played for roughly 2-3 hours. No error appears when this problem happens. One thing of note is that after both instances, I logged onto my character successfully. I then tried to exit the game almost immediately and was successful, so perhaps the problem occurs after playing a significant amount of time.