Wales minister and whip Nigel Adams said the Prime Minister had made a “grave error” by reaching out to the Labour leader in the hope of finding a consensus solution to Brexit ahead of a crunch EU summit on 10 April.
Mrs May announced her dramatic move after a marathon session of Cabinet at which a number of ministers are understood to have argued for a no-deal EU withdrawal next week.
In his letter, Mr Adams said the Government faced two “great challenges” of delivering “the Brexit the people voted for” and preventing a Corbyn premiership.
And he added: “Sadly, I fear that we are now at risk of simultaneously failing in both.”
The Selby and Ainsty MP said: “Legitimising and turning to Jeremy Corbyn to assist you at this crucial stage, rather than being bold, is a grave error.
“It is clear that we will now end up in the customs union. That is not the Brexit my constituents were promised and it is contrary to the pledge we made in our manifesto. It makes no sense to leave the EU and to have a situation where our trade policy and much of our law is made in Brussels with no say for the UK.”
Mr Adams, who was made a whip in January last year and promoted to the Wales Office in November, said he continued to believe that no deal was better than a bad deal.
He is the 35th member of the Government to resign in the past 12 months.
“It now seems that you and your Cabinet have decided that a deal cooked up with a Marxist who has never once in his political life put British interests first is better than no deal, “ he told the PM. “I profoundly disagree with this approach.”
Mr Adams’s resignation is the first since Tuesday’s Cabinet meeting and came shortly after Brexit secretary Stephen Barclay said a softer Brexit was the inevitable result of the political balance in the Commons.
Referring to a softer Brexit scenario, Mr Barclay told BBC Radio 4’s Today programme: “It’s undesirable, but it’s the remorseless logic of the numbers of the House of Commons.
“The Prime Minister’s deal won’t go through and no deal in law is taken off the table, then the consequence of that is either a soft Brexit or no Brexit at all.”
The Prime Minister’s offer to meet the Labour leader to try and end the impasse over EU withdrawal sparked fury among Tory Eurosceptics.
But Mr Barclay blamed hardliners in Tory ranks for the situation.
He said: “It’s regrettable that what we have been saying for several months now is coming to pass, but that is the remorseless logic of not backing the Prime Minister’s deal.
“Because the alternative then is to have to seek votes from the opposition benches because 35 of my own colleagues would not support the Prime Minister’s deal.”
The comments came after Mrs May said she will seek an Article 50 extension beyond next week to allow negotiations with the Labour leader aimed at ensuring the UK leaves the European Union “in a timely and orderly way”.
Mr Corbyn said he would be “very happy” to meet the Prime Minister in a bid to offer “certainty and security” to the British people - but Tory Brexiteers have reacted with anger.