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The bill, if passed, will give Prime Minister Theresa May the authorisation to trigger Article 50 - the process by which Britain leaves the 28-member bloc and the two-year negotiation period starts.
Government officials say MPs will be given time to debate and vote on the bill as early as next week.
Ministers had hoped to trigger Brexit without first consulting Parliament, but the government was forced to publish the bill after losing a Supreme Court case on the issue on Tuesday.
Sources told Business Insider that the government is now keen to rush the legislation through both houses of Parliament within weeks in order to pass it well before May's self-imposed deadline of the end of March.
The bill is likely to be passed with an overwhelming majority after Labour signalled that they would not seek to block it.
However, opposition parties are set to submit dozens of amendments to the bill in an attempt to shape the government's negotiation strategy with the EU.
Labour plans to push the government to make commitments on Britain's access to Europe's Single Market as well as on workers' rights and environmental and social protections.
The SNP are also set to submit up to 50 amendments in an attempt to prevent a so-called "hard Brexit" from the EU.
The Liberal Democrats have said they will vote against the bill unless the government commits to holding a second referendum on the terms of Britain's final Brexit deal.
May on Wednesday committed to publishing a full white paper setting out the government's plans for Brexit.
Ministers had initially resisted doing so but changed course following demands from Conservative backbenchers. No date has yet been announced for its publication.