In 2008, the USPS processed 203 billion pieces of mail.
Most mail is represented by a 1 ounce envelope; however, packages and other larger items are mailed frequently, so we can assume that the average weight of a piece of mail is more than 1 ounce.
Using 2 ounces, 203 billion pieces will weigh about 406 billion ounces.
There are 16 ounces in a pound, so 406 billion oz / 16 oz/lb = 25.375 billion pounds of mail per year.