Theresa May is to ask Brussels for only a short delay to Brexit.
Confirming a short extension will be requested, a Number 10 source said: "PM won't be asking for a long extension.
"There is a case for giving Parliament a bit more time to agree a way forward, but the people of this country have been waiting nearly three years now.
"They are fed up with Parliament's failure to take a decision and the PM shares their frustration."
The development comes exactly 1,000 days after the referendum of June 23 2016, which delivered a 52%-48% majority for leaving the EU.
It will dismay hardline Leavers still hoping for a no-deal "clean Brexit" next Friday.
In a fractious Cabinet discussion of the planned letter on Tuesday, Leave-backing Leader of the House Andrea Leadsom is understood to have said: "This used to be the Cabinet that would deliver Brexit and now, from what I'm hearing, it's not."
And US President Donald Trump's son said Mrs May should "honour" her promise to take Britain out of the EU next Friday.
In an article in the Daily Telegraph, Donald Trump Jr accused Mrs May of ignoring his father's advice and claimed that "elites control London from Brussels" and "democracy in the UK is all but dead".
The EU's chief negotiator Michel Barnier on Tuesday warned that Brussels will want a good reason to grant a long delay to Brexit.
In an apparent reference to options like a second referendum, general election or cross-party consensus, he said that a lengthy extension "needs to be linked to something new ... a new event or a new political process".
Any request for extra time is subject to unanimous approval by leaders of the remaining 27 EU states at a meeting in Brussels on Thursday.
And senior MEP Sophie in't Veld, deputy to European Parliament Brexit co-ordinator Guy Verhofstadt, warned that any extension will come with "conditions attached".
Speaking in Brussels on Tuesday, Mr Barnier said: "It is our duty to ask whether this extension would be useful because an extension will be something which would extend uncertainty and uncertainty costs."
Irish foreign minister Simon Coveney told reporters: "If there is going to be a request for a long extension of Article 50 by the UK then there will need to be a very persuasive plan to go with that to explain why that's needed and how they will use the time to conclude the outstanding issues that haven't been able to be agreed in London in the context of the Brexit process."
If the EU agrees an extension, MPs and peers will have to pass a statutory instrument within days to remove the date of March 29 from Brexit legislation.
MPs will be given an opportunity in the House of Commons on Monday to debate how the process should go forward.
And Mrs May is expected to seek a way of staging a third "meaningful vote" on her Withdrawal Agreement next week, despite Speaker John Bercow's ruling that it must be substantially changed before being put before MPs again.
Downing Street sources dismissed reports that the vote, known in Westminster as MV3, is slated to take place on the eve of Brexit on March 28, insisting that no date has been decided.
Talks are expected to continue with the DUP ahead of any vote in the hope of overturning its defeat by a margin of 149 last week.
But it is understood that the Government is still far from any agreement with the Northern Irish party that would allow its 10 MPs to back the PM's deal.
Jeremy Corbyn indicated that he sees Monday as the point for Labour to mount a challenge to the Government's approach.
Speaking after talks with other opposition parties and Labour backbenchers on the way forward, Mr Corbyn said: "If the Government can't get a majority for its way on Monday, then I think that's the time to challenge this Government.
"The reality is that this Government has lost its authority, doesn't enjoy the confidence of the House, can't get anything through.
"Surely that is the time to step aside and let the people decide in a people's vote that's called a general election."
He added: "I hope that on Monday the House will come together and support some sensible alternatives that can be negotiated during an extension period with the EU."