Sheesh! How often do I have to say it ? A millennium is a thousand contiguous years: any thousand years. One can start a millennium any time, including 2000 January 1st.
The difficulty comes when people want to identify the millennium of the 2000s (which starts on 2000 January 1) with the 3rd millennium A.D. (which starts on 2001 January 1). Both are millennia, but it is mathematically impossible to consider them exactly the same. It's true, though, that most people who speak of "the" millennium seem to be making this identification. Their words aren't technically wrong, since "the new millennium" of the 2000s begins on 2000 January 1. They could speak accurately in precisely these words. But the context usually betrays that they make the false identification of the millennium of the 2000s with the 3rd millennium A.D. or the century of the twenty-hundreds (2000-2099) with the 21st century A.D. (2001-2100).
T.