Bangladesh, Sept. 14 -- American citizenship should never be something wealthy foreign nationals can purchase through a travel package, a hospital appointment and a carefully scripted story for immigration officers. Yet that is precisely what the growing problem of birth tourism has made possible.

The practice is simple. A foreign national travels to the United States shortly before giving birth, delivers the child on American soil and then returns home. Under the longstanding interpretation of birthright citizenship, the child generally becomes a US citizen.

Around that principle, an entire industry has emerged.

Businesses advertise services that can include arranging travel, housing, hospital care and transportation. Some provide as...