Contrary to all the hype, the start of the next century/millennium is on January 1, 2001. The calendar started at year one, not zero. It didn't start at zero, because at the time, the concept of zero did not exist in Western civilization. (Try finding a Roman Numeral zero. :-) That means the first millennium included the years 1 through 1000, the second millennium the years 1001 through 2000. Yes, the rollover from 1999 to 2000 looks great, on paper, but it is the LAST year of the current century/millennium.
Actually, if you believe some of the scholars, the birth in question was four years earlier. So, when does the actual third millennium start? I certainly do not know, other than it isn't January 1, 2000. Based on the current calendar (used by a majority, but not by everyone), it is January 1, 2001.
I expect that knowing when the millennium really starts is not going to stop you from partying on December 31, 1999. Go ahead, celebrate all those zeros rolling over in the year, but consider the year 2000 as a year long party to the start of the next millenium (Greenwich does :-).
Here are links to offical sites that also say that 2001 is it:
your computer's clock.