Switzerland's oldest bank is to close after pleading guilty to helping some of the US's richest people evade paying taxes on at least $1.2bn (£750m) which was hidden in secret offshore accounts.

Wegelin, which was founded in 1741, said it would "cease to operate as a bank" after it admitted it had allowed 100 US taxpayers to hide their money.

The bank agreed to pay $57.8m in fines and restitution to the US authorities after admitting to conspiracy charges related to helping US taxpayers living overseas evade payments to the Internal Revenue Service (IRS) for almost a decade.

Otto Bruderer, a managing partner of the bank, told a New York court: "Wegelin was aware that this conduct was wrong … From about 2002 through to about 2010, Wegelin agreed with certain US taxpayers to evade the US tax obligations of these US taxpayer clients, who filed false tax returns with the IRS."

The bank, which started business 35 years before the US declaration of independence, released a statement confirming its closure late on Thursday. "Once the matter is finally concluded, Wegelin will cease to operate as a bank," it said from its headquarters in the small town of St Gallen, near the border with Austria and Liechtenstein.

It is the first foreign bank to close since the US authorities began a crackdown on those helping Americans dodge taxes.

US authorities said Wegelin had wooed American clients fleeing Switzerland's biggest bank, UBS, after it admitted in 2008 to helping Americans evade tax, paid a $780m fine and handed over information on more than 4,450 accounts.

US prosecutor Preet Bharara said Wegelin became a haven for US tax evaders by hiding their money in secret offshore accounts.

Bharara, who was described as the man "busting Wall Street" on the cover of Time magazine, said Wegelin "wilfully and aggressively jumped in to fill the void that was left when other Swiss banks abandoned the practice due to pressure from US law enforcement".

He said Wegelin's closure and guilty plea was a watershed moment in US efforts to crack down on individuals and banks "engaging in unlawful conduct that deprives the US treasury of billions of dollars of tax revenue".

"There is no excuse for wealthy Americans flouting their responsibilities as citizens of this great country to pay their taxes, and there is no excuse for foreign financial institutions helping them to do so," he said.

It was alleged that Wegelin's scheme involved its bankers opening secret accounts for US clients under code names and setting up sham entities to avoid detection in various tax havens, including Panama and Liechtenstein.

It is not yet known if the bank will be forced to hand over the names of US clients who held secret accounts at the bank. Until this week it had refused to appear in court to answer the charges, leading US district judge Jed Rakoff to declare it a fugitive from justice.

In court papers, Bruderer said Wegelin "believed it would not be prosecuted in the United States for this conduct because it had no branches or offices in the United States and because of its understanding that it acted in accordance with, and not in violation of, Swiss law and that such conduct was common in the Swiss banking industry".

Although Wegelin will cease to exist, the bank's partners sold its non-American client accounts to the Austrian bank Raiffeisen just before its indictment last January. Rakoff described the sale as a "fraud upon a fraud".

William Sharp, a tax lawyer in Tampa, Florida, who advises many US clients of Swiss banks, described Wegelin's decision to change its plea to guilty as shocking and said it should serve as a wake-up call to foreign banks helping US clients to make sure they comply with US laws.

SwissPrivateBank.com, a website Wegelin used to attract English-speaking clients, boasted that: "Swiss bank secrecy is not lifted for tax evasion … Neither the Swiss government nor any other government can obtain information about your bank account."

Three of Wegelin's executives – Michael Berlinka, Urs Frei and Roger Keller – who were also indicted last year, are expected to avoid the charges because the extradition treaty between Switzerland and the US does not include tax crimes.

US authorities are investigating similar allegations about at least 11 other banks, including Credit Suisse.