I will spare everyone the mathematical aspects of this, but below is an excerpt of the article "How Much Does the Internet Weigh? We put cyberspace on the scale" by Stephen Cass as published in the June 2007 issue of Discover:
At last, after much scribbling (and perhaps a little cursing), we had our answer: The weight of the Internet adds up to just about 0.2 millionths of an ounce." [page 43]So what does 0.2 millionths of an ounce consist of? According to the same article which quoted Clifford Holliday, author of Internet Growth 2006, 75% of the cyberspace is filled by file sharing activities. Of this, 59% is attributed to people swapping video files, 33% music tracks, and a meager 9% is used for e-mails.
So now we know.