Fill out Form DS-82, available on the U.S. Department of State website. This is the basic form for applying for a U.S. passport by mail, and is not specific to a duplicate passport. This form will ask for basic information, such as name, mailing address, Social Security number, place of birth, emergency contacts, and any upcoming travel plans. Print off a copy of the form. While the form is available to submit online, you'll need to either mail or bring in person your application for a second passport.
Write a letter explaining your need for a duplicate passport. This does not need to be long; one paragraph is ideal. Travel plans that require a duplicate passport help your case, so include these if possible, or provide a history of trips where a duplicate passport would have been needed. Be as specific as possible when discussing why you should be able to obtain a duplicate passport.
Submit Form DS-82, two passport photos, any passport fees, and the letter explaining your need for a duplicate passport. As of October 2009, the fee for a duplicate passport was $135. It will take four to six weeks to get a response on whether your duplicate passport request has been approved.