Prime Minister Theresa May's spokesman had earlier confirmed the date for the official triggering of Article 50 as 29 March.
Notification comes 279 days after the referendum of June 23 previous year delivered a 52 per cent to 48 per cent majority in favour of withdrawal.
Dr White said the Bills, dealing with issues such as immigration procedures, would be on top of the Great Repeal Bill, which will incorporate European Union law into British law.
A spokesman for the European Commission said: "Everything is ready on this side".
We have always said we meant to do so before the end of March.
Mrs May will address MPs in a statement to the House of Commons following her regular weekly session of Prime Minister's Questions on March 29.
The prime minister has said that "now is not the time", and is expected to visit Scotland as part of a pre-Brexit tour of Britain that began on Monday in Wales and will also take in Northern Ireland.
He said the Government wanted to secure "a new, positive partnership between the United Kingdom and our friends and allies in the European Union". Such a transitional deal would allay fears about an impending cliff edge in March 2019, when Britain is set to crash out of the single market and face the EU's steep external tariffs on goods ranging from meat to cars.
In response to the news, European Union president Donald Tusk said he would have draft negotiating guidelines ready within 48 hours of receiving the letter.
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Negotiations will begin to outline a UK Brexit deal, which could include trade deals - although this might be handled separately.
Secretary of State for Exiting the European Union, David Davis, said: 'Last June, the people of the United Kingdom made the historic decision to leave the EU.
The prime minister also says she wants a "phased period of implementation" of a new relationship with the European Union to give businesses time to plan.
It is not quite accurate to say nothing much is going to happen after article 50 is triggered next week, but the huge excitement built up around "the date" is likely to prove misplaced.
The developments come as an independent think tank warned today that up to 15 new bills could be needed to deliver Brexit.
The Scottish National Party's Europe spokesperson at Westminster, Stephen Gethins, said: "Today's announcement. shatters beyond fix any notion or position that the Prime Minister is seeking a UK-wide agreement".
The European Union, meanwhile, is fully prepared for Brexit negotiations after Britain announced that it would trigger its divorce on March 29, the European Commission said on Monday.
Later this week, EU leaders will gather in Rome to mark the 60th anniversary of the Treaties of Rome, which established the European Economic Community - the initial forerunner to the EU. Mrs May is not attending the event.