I think Priceguy’s just going to say “to hell with it.”
But, while Ebola Gay’s strictly correct, there are conventions. Every single digital clock or watch that I’ve even seen shows the minutes in this sequence: 11:59pm; 12:00am; and 12:01am. Similarly pm sequences thus: 11:59am; 12:00pm; 12:01pm. The Windows and Macintosh clocks exhibit this behavior, too. This makes sense sequentially. The 00 minutes is the first minute, because 00 minutes 1 second is within the domain of the first minute. Oddly, this is just the opposite of my Y2K belief (I celebrated the 21st century a year later than most people).
Now for midnight, if I were to say, “tonight at midnight” I would mean tonight just after 11:59pm. This is really a colloqialism on my part, since I would also say next Tueday at midnight I would also mean just after 11:59 on Tuesday. But were I to say 12:00am on Tuesday, I would mean the midnight between Mon. and Tues. Actually I know this causes confusion depending on whom I’m talking to, so I’m usually more verbose and then say something like at 12:00 am Monday night Tuesday morning just to be clear.