Switzerland made a desperate decision on Wednesday to save its battered private banking sector by allowing Swiss banks to cooperate with a broad tax evasion probe by U.S. investigators. It is likely to unleash a flood of fresh disclosures to the IRS by American taxpayers, lawyers say.

Such disclosures could trigger hefty fines that can reach a multiple of what an account holds. To stand a chance of reducing those fines, Americans with hidden Swiss accounts “should run, not walk, to jump in line with an IRS voluntary disclosure in light of this move by the Swiss government,” says Josh Ungerman, a tax and estates lawyer in Dallas and a former IRS prosecutor.

Lawrence Horn, a tax and business crimes lawyer in Newark, N.J. and a former federal prosecutor, says he expects “at least 10,000” American taxpayers to come forward in the next 12 months or so.

The Swiss decision has two key points: Over the course of 12 months, banks will be allowed to turn over to American authorities general statistical data on their work with American clients. More significantly, the U.S. can seek concrete account details — with names of taxpayers — if the U.S. Senate gets its act together and passes a double-taxation treaty already green-lighted by Bern.

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The latter all but guarantees that a client’s name will be disclosed. But so, possibly, does the former: The IRS already has a trove of information from previous disclosures, including names of bankers, and prosecutors are working with authorities in other tax haven countries, notably Liechtenstein, to scare up details on Swiss banks and their clients. Around a dozen banks have been ensnared in the U.S. investigation, including UBS (UBS), Credit Suisse (CS), HSBC’s Swiss arm, Julius Baer and two cantonal, or regional, banks.

Wednesday’s decision by Bern, a significant moment in the annals of Swiss secrecy and client confidentiality, awaits approval by Swiss parliament, where some factions appeared incensed by the appearance of caving in to U.S. demands. Martin Janssen, a leading Swiss finance professor at Zurich University, called the decision “blackmail by a large nation,” referring to U.S. threats to indict banks should Switzerland not agree to cooperate. Prosecutors indicted Wegelin, Switzerland’s oldest bank that once had Napoleon as a client, in 2012, and the bank pleaded guilty in January, effectively shutting its doors for good.

Still, under U.S. tax laws that make willful nondisclosure a felony, American taxpayers who are eventually outed to the IRS under the Swiss decision could face draconian fines, penalties, and criminal charges. More than five-dozen Americans have been charged in recent years for tax evasion through Swiss and other offshore banks. So it’s critical that Swiss bank account holders come forward to the IRS before the agency finds them. Unlike the U.S., Switzerland does not generally consider tax evasion to be a crime.

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Last year, the IRS revived its Offshore Voluntary Disclosure Initiative, offering tax evaders terms that were less generous compared with those under two prior programs in 2009 and 2011. Under the reopened program, entrants must file original and amended tax returns, pay back-taxes and interest for up to eight years and pay accuracy-related or delinquency penalties. The penalties, as a percentage of assets, range from as low as 5% to as high as 27.5%, depending on the amount of assets they hold overseas. Taxpayers caught by the IRS before entering the program can face penalties equal to 50% of the highest value in their accounts for each year of violation, a calculus that can make fines outweigh assets.

Two prior disclosure initiatives, in 2009 and 2011, brought in tens of thousands of taxpayers and $4.4 billion, according to IRS data. In all, some 38,000 taxpayers have come forward and paid $5.5 billion. Others have tried a “quiet disclosure” — filing returns, paying taxes, and hoping the IRS doesn’t notice or levy fines. Horn says a subset of tax evaders “are people who are resistant and calculated, who think they’re different and will take the risk” of continuing to hide their accounts.

“When it comes to U.S. taxpayers and Swiss bank secrecy, Elvis has not only left the building, he is not coming back,” says Robert Katzberg, a white-collar criminal defense lawyer in New York with Swiss bank clients. “A shockingly large number of U.S. taxpayers with undisclosed accounts have failed to realize this, but maybe now they will.”