KANSAS CITY, Mo. -- While U.S. airlines downsize and scrimp on amenities, one carrier is offering its passengers leather seats, ample legroom and free food. But frequent fliers probably don't want a ticket on what may be the fastest growing "airline" serving Central America.

This carrier is run by U.S. Immigration and Customs Enforcement, the federal agency responsible for finding and deporting undocumented immigrants. A crackdown on illegal immigration has led to a spike in deportations and the creation of a de facto airline...