I would have thought it's AM for the whole of the 12th hour. So 12:00 AM, 12:01 AM, 12:59 AM, 1:00 PM (hour one, past the hour of noon). An hour being a time span of 60 minutes, not an instant!

In the "correct" usage, you have an illogical jump between 11:59 AM and 12:00 PM.