The British Cabinet – including Mr Davis – signed off on the plan last Friday
Dominic Raab has been announced as the new UK Brexit Secretary David Davis.
His appointment comes after David Davis’ shock resignation overnight.
In a letter to the British Prime minister, Mr Davis warned that her latest Brexit proposals were ‘unworkable.’
He said he believed it now looked "less and less likely" the party would deliver on the Brexit result and the Tory commitment to leave the customs union and single market.
"The general direction of policy will leave us in, at best, a weak negotiating position – and possibly an inescapable one," he wrote in his resignation letter.
Mr Davis's junior minister Steve Baker has also resigned.
Mr Raab was previously a housing minister under Communities Secretary James Brokenshire.
He is known as a Brexit supporter, however he would be viewed as more liberal than other elements of the Conservative party.
A 10 Downing Street statement said: "The Queen has been pleased to approve the appointment of Dominic Raab MP as Secretary of State for Exiting the European Union.”
"This follows the departure from government of the Rt Hon David Davis MP."
The plan, agreed by British cabinet ministers on Friday, will set up a new EU / UK customs area, although Britain would be entitled to apply different tariffs than the EU.
It would keep the UK aligned with EU rule for goods, while it moves away in terms of services.
"Yesterday the Cabinet agreed our Brexit #DealForBritain. Here are 12 key principles that we will use in our EU negotiations as we continue along the road to Brexit." - PM @Theresa_May pic.twitter.com/6GuAT5dKwL— UK Prime Minister (@10DowningStreet) July 7, 2018
Taoiseach Leo Varadkar has said he is looking forward to examining the detail of the plan – due to be published in a White Paper later this week.
He said he was more optimistic than ever that a deal could be completed by the end of the year – but warned that concerns remain over the workability of the customs proposals and the overall effect on the integrity of the single market.
Good phone call with PM May just there. Welcomed proposals from British Government which can input into talks on the future relationship. Await detail in white paper late next week before agreeing an EU position.— Leo Varadkar (@campaignforleo) July 7, 2018
Yesterday, the Tánaiste Simon Coveney welcomed what he called a softening of the UK's approach.
“I think the only way you can interpret the political developments out of Britain in the past number of days is as a softening in the approach,” he said.
“Bu I also think it is based on realism. British business wants to trade into the EU.
“The EU Single Market is by far the most important market for British manufacturers and so on – just like they are a hugely important market for us.”
Theresa May is due to outline the approach in parliament later, before a potentially stormy meeting with her own MPs.
Writing to Mr Davis after receiving his resignation, she warned: "I do not agree with your characterisation of the policy we agreed at Cabinet on Friday."
She set out 12 points explaining "how we will deliver on the result of the referendum and the commitments we made in our manifesto."
They include "ending free movement," "a new business-friendly customs model," "no more sending vast sums of money each year to the EU" and "no hard border between Northern Ireland and Ireland."
The PM adds: "At Chequers on Friday, we as the Cabinet agreed a comprehensive and detailed proposal which provides a precise, responsible, and credible basis for progressing our negotiations towards a new relationship between the UK and the EU after we leave in March."
Mr Davis had attacked the plans, saying they will make "the supposed control by Parliament illusory rather than real."
"I am also unpersuaded that our negotiating approach will not just lead to further demands for concession," he tells the prime minister.
He ends by saying that Mrs May needs "an enthusiastic believer in your approach, and not merely a reluctant conscript."
The country’s Foreign Secretary Boris Johnson reportedly described defending the Brexit plans as like "polishing a turd" during the Chequers summit, before eventually falling into line behind the prime minister.
After Mr Davis’ resignation, leading Brexiteer Jacob Rees-Mogg said: "This is very important. It raises the most serious questions about the PM's ideas.
"If the Brexit Secretary cannot support them they cannot be very good proposals. It was an attempt to bounce the cabinet. It was a serious mistake."
He added: "It is not her [Theresa May's] finest night in politics."
A leadership contest would be triggered if 48 Conservative MPs formally submit letters, and some have already reportedly been sent to the Tories' backbench 1922 Committee.
In her appearance in the Commons later, Mrs May will admit the cabinet have had "robust views" but will insist she has listened to "every possible idea" and that "this is the right Brexit."
She will tell MPs the proposals "will deliver on the democratic decision of the British people."