iPhone 5 Dies At 100%?

Try a restore: http://support.apple.com/kb/HT1414

Restore as new, not from a backup. If that does not work, you probably need to have the battery replaced, so make an appointment at the Apple Store: http://apple.com/retail/geniusbar.

Read the article:

Before you restore your device

1. Make sure you’re using the latest version of iTunes.
2. Use iTunes to back up your device and transfer and sync any additional content to your computer.
3. Turn off Find My iPhone in Settings > iCloud on your device to disable Activation Lock.

Yes, as long as you have a backup and you do not sync after restoring as New. To be safe launch iTunes, go to iTunes Preferences via the menu, Devices tab, and check the box to NOT automatically sync devices when connected.

I also suggest 2 backups, one on your computer, and another in iCloud (Settings/icloud).

No I'm not. I'm telling you to make two backups (for extra security) then restore as New, but do not sync. That way a sync will not overwrite your backup.

1. Restore as new.
2. See if that solves the problem.
3. If it does not solve the problem you need a new battery.
4. If it does solve the problem restore the backup.
5. If the problem returns you have bad data in the backup.

First, Restore Backup. Then Sync. This will put your phone the way it was before you started.

NO, you are not getting the message. Use RESTORE IPHONE FIRST!!!!!!!!

When asked choose to set up as NEW.

Does that solve the problem?

NO, the problem is not solved: You have a bad battery. Take it to an Apple store and they will replace the battery.

YES, the problem has gone away: NOW use Restore Backup

Does the problem return?

NO: your phone is fixed. Now click Sync to restore your music, videos and apps

YES, the problem returns: Your backup contains corrupt data. You will have to restore as NEW again and do not restore your backup. You will have to manually reinstall your apps and sync your music. You will lose app data and settings.