A period of 1000 years.Since there was no year 0, the years 1-1000 AD correspond to the first millennium, 1001-2000 to the second, etc. The third millennium therefore began on January 1, 2001. The situation is summed up succinctly in the following editorial from the Times of London on December 26, 1799 (NPR Morning Edition), which discusses the analogous issue of when the new century starts.

"We have uniformly rejected all letters and declined all discussion on the question of when the present century ends, as it is one of the most absurd that can engage the pubic attention, and we are astonished to find it has been the subject of so much dispute, since it appears plain the present century will not terminate until January the first, 1801, unless it can be made out that 99 are 100. It is a silly, childish discussion, and only exposes the want of brains in those who maintain a contrary opinion to that we have stated."

However, rather than wasting time arguing about whether 2000 or 2001 is the start of the "new" millennium, we can simply note that the year 2000 corresponds to an odometer-like "rolling over" of the digits of the year, a very rare event which is clearly a calendrically significant!

Because 2000, 4000, 6000, etc. are leap years and 1000, 3000, 5000, etc. are not, the number of leap days in each millennium alternates between 242 and 243, with the first, third, etc. millennia (i.e., 1-1000, 2001-3000, etc.) having 242 leap days, and the second, fourth, etc. (i.e., 1001-2000, 3001-4000, etc.) having 243 leap days.