The New York Times, for example, didn't do a special story on January 1, 1900, whereas on January 1, 1901, it ran a front-page one with the following headline:
TWENTIETH CENTURY'S TRIUMPHANT ENTRY; Welcomed by New York with Tumultuous Rejoicing.
And a first paragraph that began:
"The Century is dead; long live the Century! Yesterday was the Nineteenth, today the Twentieth. Some time last night the one died and the other was born. Here in New York the event took place at midnight."
By the way, today's Times has an article about 1890s men's fashion coming back in style; the link may work only if you've previously been able to access Times articles on your computer:
http://www.nytimes.com/2009/11/12/fashion/12CODES.html