Is it correct to say "12:00am"?

I've read in various places the first minute of a day described as 12:00am. Now, whilst I personally prefer to use 24h clock notation and therefore don't have this problem as I can simply describe this minute as 00:00, how is one supposed to express this time using am/pm notation? Is 00:00am common (strange because there is no 00:00pm), or is the least-bad way to say 12:00am (strange IMHO because it seems to imply that the previous hour was 11:00am when it was actually 11:00pm)?

The Wikipedia article 12 Hour Clock is worth quoting at length:

Confusion at noon and midnight

It is not always clear what times "12:00 a.m." and "12:00 p.m." denote. From the Latin words meridies (midday), ante (before) and post (after), the term ante meridiem (a.m.) means before midday and post meridiem (p.m.) means after midday. Since strictly speaking "noon" (midday) is neither before or after itself, the terms a.m. and p.m. do not apply. However, since 12:01 p.m. is after noon, it is common to extend this usage for 12:00 p.m. to denote noon. That leaves 12:00 a.m. to be used for midnight at the beginning of the day, continuing to 12.01 a.m. that same day.

However, because practical confusion is still possible, some style guides recommend replacing "12:00 p.m." with "12:00 noon" and "12:00 a.m." with "12:00 midnight".

The American Heritage Dictionary of the English Language (Fourth Edition, 2000) has a usage note on this topic: "Strictly speaking, 12 a.m. denotes midnight, and 12 p.m. denotes noon, but there is sufficient confusion over these uses to make it advisable to use 12 noon and 12 midnight where clarity is required."

Many U.S. style guides, and NIST's "Frequently asked questions (FAQ)" web page, recommend that it is clearest if one refers to "noon" or "12:00 noon" and "midnight" or "12:00 midnight" (rather than to "12:00 p.m." and "12:00 a.m."). Some other style guides suggest "12:00 n" for noon and "12:00 m" for midnight.

The Canadian Press Stylebook (11th Edition, 1999, page 288) says, "write noon or midnight, not 12 noon or 12 midnight." Phrases such as "12 a.m." and "12 p.m." are not mentioned at all.

The use of "12:00 midnight" or "midnight" is still problematic because it does not distinguish between the midnight at the start of a particular day and the midnight at its end. To avoid confusion and error, some U.S. style guides recommend either clarifying "midnight" with other context clues, or not referring to midnight at all. For an example of the latter method, "midnight" is replaced with "11:59 p.m." for the end of a day or "12:01 a.m." for the start of the next day. That has become common in the United States in legal contracts and for airplane, bus, or train schedules, though some schedules use other conventions.

My advice is to use "12:00 noon" instead of "12:00 p.m." and "12:00 midnight" instead of "12:00 a.m." Realize that readers may be confused which day 12:00 midnight belongs to, so make sure context makes it clear.

12:00am is correct.
The clock goes from 12:00am (midnight) through 1:00am, 2:00am to 11:59am and then to 12:00pm (noon) through 1:00pm to 11:59pm and repeats.

It seems no more strange to go from 12:59 to 1:00 than it does to go from 11:59pm to 12:00am