With few exceptions, a divorced parent cannot leave the United States with her child unless she first obtains permission from the child's other parent. This is true even if the parent making the trip is the custodial parent. According to a 2007 report by CNN, roughly 800,000 children are reported missing each year -- and about one-fourth of these abductions are committed by family members. While there is no way to prevent kidnappings altogether, requiring an individual to obtain permission from the child's other parent before traveling abroad makes kidnapping the child more difficult.
You and your child must both have a passport before you can travel to another country by air. If you are traveling to contiguous territory, such as Mexico or Canada, you must prove your child's citizenship by providing either your child's original birth certificate or a certified copy. You must also provide a written statement, signed by your former spouse, stating that your child has permission to leave the country.
If you plan to travel to non-contiguous territory with your child, you and your child must both have a passport. The U.S. Department of State notes that minors under age 16 must have permission from both parents to obtain a passport. You must apply for a minor's passport in person. Your former spouse can either go with you to fill out the passport application, or provide you with a written, signed and notarized statement giving the U.S. Department of State permission to issue your child a passport.
A parent with sole custody does have the legal right to make decisions about his child without seeking input from the child's other parent. This includes the right to apply for a passport for his child and leave the country. You must prove that you have sole custody by providing the U.S. Department of state with a copy of the court-ordered custody arrangement before your child can obtain a passport. If you are traveling to contiguous territory, you must bring a copy of the court's decision with you when you cross the border.