I rather like the approach taken by the Calendars app (available on iOS).
By default, it shows this view (focus on the bottom parts of the screenshots):
It does not allow for single minute precision. However, for a lot of applications, I'd imagine that being acceptable. I rather like the division of the time that was chosen. Rather than AM/PM ranges, the upper bar contains the day-time hours, and the lower bar contains the night-time hours. The only weird result of this, is the placement of the 0:00 to 07:00. It right after the 20:00 to 23:00 range, but it represents the time before it. I guess a three-bar variation (with empty parts in them) would be clearer, but not as visually appealing.
On clicking the Date button (or on the All-day button), the view changes to this:
This is of course all in an iOS app, not a website. But you might use a similar approach online, I think.