''The Rio Rico case is the only known instance of a private diversion of the river,'' said Manuel R. Ybarra, secretary to the American section of the United States-Mexico International Boundary and Water Commission. ''The company was taken to court and fined, but by then the damage had already been done.'' Founded During Prohibition

For many years that damage went all but unnoticed. The United States never formally relinquished title to the land, but it was administered as part of Mexico, even after this town was founded in 1929 as a lure to Texans seeking drink and entertainment denied them by Prohibition.

In 1972, however, Homero Cantu Trevino filed suit to prevent the United States Immigration and Naturalization Service from deporting him from Texas to Mexico. He argued that, having been born here in 1935, he was a citizen of the United States, not Mexico.

The Justice Department fought the claim in court. In 1976, a judge found in the Government's favor. A year later, though, an appeals court ruled that because the 1906 cutoff was unauthorized, anyone born in Rio Rico before 1972 was entitled to American citizenship.

While the Government argued that lack of American jurisdiction should deprive Mr. Cantu of citizenship, it undercut its case when the State Department formally transferred the Rio Rico tract to Mexico in the 1970's. It also appeared to acknowledge Mr. Cantu's citizenship by paying back taxes to Texas on the federally owned land for the years after 1906. A Migration to the U.S. With the court decision, virtually every able-bodied man born here left to work in the United States. Today, Rio Ricans can be found from Florida to Hawaii. The only people still living here are the elderly, children born after 1972, adult newcomers and those few unable to prove they were born here.