The crackdown eventually forced Wegelin, Switzerland's oldest private bank, to close after pleading guilty in 2013 to helping wealthy U.S. citizens avoid paying taxes and agreeing to pay $74 million.
The Wegelin case came after UBS AG became the first bank to settle with the U.S. government, agreeing in 2009 to pay $780 million and entering into a deferred prosecution agreement.
Credit Suisse Group AG in 2014 pleaded guilty to conspiracy to aid and assist U.S. taxpayers in filing false returns as part of a more than $2.5 billion settlement with various authorities.
Sentencing of Amrein, who faces up to five years in prison, was set for July 1.
The case is U.S. v. Amrein, U.S. District Court, Southern District of New York, No. 13-cr-00972.