Afidavite

Check with your local courthouse, but in Colorado you have to put an ad in the newspaper at the fathers last known address (either address you know, phone book, etc.) for 30 days (I believe - may be 60 now) stating you are "looking" for him for legal reasons regarding the child. You need to keep daily copies of these ads and show them to the court with no response and fathers rights are taken away - basically considered abandonment and you have 100% custody and can get the child a passport.

If you find an easier way please let me know....

If your child is under the age of 15 - you will need to have the info that mma_mom stated above. If your child is over 15 years old, only the mother is needed to sign for the passport. The government has some detailed information online regarding passports. Good luck!