Mrs May said she was “sorry” to receive Mrs Leadsom’s resignation letter, writing in reply that she was “grateful for the support you have given over the last three years” in working to deliver Brexit.
But the British Prime Minister said she disagreed with the assessment Mrs Leadsom gave about the UK Government’s approach now.
She said: “I do not agree with you that the deal which we have negotiated with the European Union means that the United Kingdom will not become a sovereign country.
“If the deal is passed, the UK will leave the European Union. We will leave its Common Agricultural Policy and its Common Fisheries Policy.
“We will end the jurisdiction of the European Court of Justice and make our own laws in Westminster and our devolved parliaments.
“We will stop sending vast annual sums of taxpayers’ money to the EU, and spend it on our priorities instead – such as our National Health Service and our children’s schools.”
Mrs May said she agreed that a second referendum would be divisive, but said the Government was not proposing to hold one.
She wrote: “I continue to believe that the arguments against a second referendum are strong and compelling, and will continue to oppose one.”
The PM said she also did not recognise what Mrs Leadsom said about decision-making in Government.
“Indeed, through your own work chairing the Parliamentary Business and Legislation committee of Cabinet, you have been an important part of ensuring that Brexit-related legislative proposals are properly scrutinised, and the whole Cabinet have listened to the assessments you have given at Cabinet meetings,” Mrs May said.
“I have valued the frank and productive discussions we have had over the last three years.
“As you say, there are important elections tomorrow, and many Conservatives are working in a challenging environment to support our excellent candidates. I am sorry to lose someone of your passion, drive and sincerity from HM Government in this way.”
The leader of the House of Commons Andrea Leadsom has resigned, saying in a letter to Theresa May “I no longer believe that our approach will deliver on the referendum result”.
It is with great regret and a heavy heart that I have decided to resign from the Government. pic.twitter.com/f2SOXkaqmH— Andrea Leadsom MP (@andrealeadsom) May 22, 2019
Mrs Leadsom, a former Tory leadership candidate in 2016, outlined four reasons for her decision in her letter.
She said: “1. I do not believe that we will be a truly sovereign United Kingdom through the deal that is now proposed;
“2. I have always maintained that a second referendum would be dangerously divisive, and I do not support the Government willingly facilitating such a concession. It would also risk undermining our Union which is something I passionately want to see strengthened;
“3. There has been such a breakdown of government processes that recent Brexit-related legislative proposals have not been properly scrutinised or approved by Cabinet members;
“4. The tolerance to those in Cabinet who have advocated policies contrary to the Government’s position has led to a complete breakdown of collective responsibility.”
Explaining the timing of her decision to resign on the eve of polling day in the European elections, Mrs Leadsom said: “I considered carefully the timing of this decision, but I cannot fulfil my duty as Leader of the House tomorrow, to announce a Bill with new elements that I fundamentally oppose.
“I fully respect the integrity, resolution and determination that you have shown during your time as Prime Minister.
“No one has wanted you to succeed more than I have, but I do now urge you to make the right decisions in the interests of the country, this Government and our party.”
The chairman of the 1922 Committee of Tory MPs, will meet the Prime Minister on Friday, sources have told the Press Association.
It comes as Graham Brady's group met to discuss the Prime Minister's future amid Mrs May's attempts to force through her Brexit deal.
Tory MP Michael Fabricant tweeted: "Executive of 1922 has made no decision. Paralysis.
"I wonder if there should be a vote of No Confidence in the 1922 Executive? Colleagues are growing impatient."
Executive of 1922 has made no decision. Paralysis.— Michael Fabricant 🇬🇧 (@Mike_Fabricant) May 22, 2019
I wonder if there should be a vote of No Confidence in the 1922 Executive? Colleagues are growing impatient.
One Tory MP described the outcome of the 1922 Committee meeting as "the can kicked down until Friday" when Sir Graham Brady will meet Theresa May.
Another MP confirmed that Mr Brady would meet Mrs May on Friday, but the chairman did not say what he would tell her.
Leaving the 1922 Committee meeting, Tory MP Steve Baker said of the timetable ahead: "The Prime Minister will be campaigning for the European elections and on Friday she will meet Sir Graham.
"Obviously they will discuss matters of interest.
"It's very much going to depend on the result of the elections and I'm sure (the media) will be able to give us a flavour.
"People are rather impatient, but equally most colleagues do appreciate this is a very difficult time for everyone."
An opinion poll from YouGov has the Conservatives in fifth place on just 7% ahead of this week's European elections, a full 30 points below the Brexit Party.
European Parliament voting intention:— Britain Elects (@britainelects) May 21, 2019
BREX: 37% (+3)
LDEM: 19% (+2)
LAB: 13% (-2)
GRN: 12% (+1)
CON: 7% (-2)
CHUK: 4% (-)
UKIP: 3% (-)
via @YouGov, 19 - 21 May
Chgs. w/ 17 May
Earlier: 'Big news expected tonight': British MPs meet to discuss forcing Theresa May to step down
In the UK, Conservative MPs are meeting to consider a rule change, which could allow another confidence vote in Theresa May.
There are also rumours of a row in her own top team over the British Prime Minister's new Brexit offer.
At least one minister has asked for a meeting with her this evening - some in her Cabinet could be prepared to tell her to go.
UK Labour MP Rosie Duffield said she has seen members of the Conservative Party "actually running along corridors" in Parliament, adding there is "big news expected" tonight.
She tweeted: "Members of Parliament from a certain Tory faction actually running along corridors.
"Lots of rumours flying around. Big news expected tonight... #EndOfDays?"
Members of Parliament from a certain Tory faction actually running along corridors. Lots of rumours flying around. Big news expected tonight....🤔 #EndOfDays ?— Rosie Duffield MP (@RosieDuffield1) May 22, 2019
Tory MP Tom Tugendhat, chairman of the Commons Foreign Affairs Committee, said he too now believed the Prime Minister must go.
Writing in the Financial Times he said that was the only way to avoid a no-deal Brexit.
"There is one last chance to get it right and leave in an orderly fashion," he wrote.
"But it is now time for Prime Minister Theresa May to go - and without delay.
"She must announce her resignation after Thursday's European elections. And the Conservative party must fast track the leadership process to replace her."
The executive of the 1922 Committee are to meet with the chief whip at 5.30pm, sources said.
Tory MPs will then join the meeting at 6pm.
Tory MP Mark Francois, leaving the meeting of the 1922 committee said: "If I were a betting man, I'd bet £50 for Help for Heroes that the second reading of the WAB will never ever happen and I invite another colleague, whoever it may be, to take my bet."