In future, investment income and capital gains arising from these assets would also be covered by a final withholding tax, at a rate of between 27 and 48 percent.
Swiss banks would pay 500 million francs ($629 million) to British tax authorities, a sum which would be offset by any voluntary payments by British taxpayers.
"Today in Zurich, British and Swiss negotiators concluded the negotiations on outstanding tax issues and initialled a tax agreement," said the ministry.
"Under this agreement, persons resident in the United Kingdom can retrospectively tax their existing banking relationships in Switzerland either by making a one-off tax payment or by disclosing their accounts," it added.
Those who choose to make an anonymous lump-sum payment would be subject to a tax rate of between 19 to 34 percent of the assets. The magnitude of the tax rate would depend on the duration of the account held in Switzerland, as well as the sum of capital.
The deal is largely similar to another accord struck between Switzerland and Germany on August 10, which also allows German taxpayers to regularise their hidden assets through a one-off lump sum tax payment at the same rates.
In addition, like the German deal, to prevent new undeclared funds from entering Swiss banks, British authorities would be able to request information on suspected tax cheats.
The number of requests made "will be in the low to mid-hundreds and not exceed 500 per year," said the finance ministry.
With Wednesday's deal, Britain has also said it does not envisage buying any stolen bank client data in order to snare tax cheats, claimed Bern.
"The two agreements with Germany and the UK show that Switzerland is serious about implementing its white money strategy," said Finance Minister Eveline Widmer-Schlumpf.
"This creates legal certainty and will strengthen the competitiveness and the reputation of Switzerland as a financial centre in the long term," she added.
The Swiss Bankers Association welcomed the deal.
"As a banker, I am especially grateful that clients have been offered a fair solution for regularising their assets," said Patrick Odier, chairman of the association.
Unconfirmed estimates of British assets hidden in Switzerland range from 59.6 billion francs to 162 billion francs.
Switzerland came under intense pressure over its banking secrecy rules following the financial crisis, as governments, reeling in debt, sought to recover taxes from assets hidden by their citizens in Swiss banks.
Facing a threat from the OECD to put it on a tax haven blacklist, Switzerland finally moved to ease banking secrecy by negotiating several deals with countries to offer cooperation to root out tax evaders.