Discover magazine did an article several years ago, called "How much does the internet weigh?". In the article, they assumed that a zero weighed nothing, and a one weighed however many electrons it took to store that value on a mempry chip. If I remember correctly, all the data on the internet weighed way less than one billionth of an ounce.
OH look, here's the article!
Forgive any misrepresentations, I didn't re-read the article, and I may have mis-remembered the facts.