International Trade Secretary Liam Fox said there is no form of customs partnership with the European Union that "could ever be acceptable" as he warned peers against thwarting the will of the people.
In a fresh blow to Theresa May's Brexit plans, the House of Lords voted to give Parliament a decisive say on the outcome of the negotiations.
Dr Fox suggested peers were using a "backdoor mechanism" to delay exit from the EU "indefinitely".
'I don't think there is a customs union that could ever be acceptable' - the possibility of some sort of customs union is 'null and void' - International Trade Secretary Liam Fox tells #r4today
-- BBC Radio 4 Today (@BBCr4today) May 1, 2018
He told BBC Radio 4's Today programme: "We can't have a situation where the clearly expressed will of the people in a referendum is thwarted by effectively procedural devices that would keep us in the EU indefinitely."
He added: "I think there is quite a big debate now about whether the unelected House can actually thwart the view of the British electorate in a referendum and what's been happening in terms of the legislation coming from the House of Commons."
Dr Fox denied that a string of parliamentary defeats being inflicted on the Government would eventually force it into accepting that the UK would have to be part of some form of customs union with the EU.
He said: "If we were in a customs union with the European Union we would have to accept what the EU negotiated in terms of market access to the UK without the UK having a voice.
"That's worse than the position in which we found ourselves today in the European Union."
"I don't think there is a customs union that could ever be acceptable," he added.
"If we are in a customs union of any sort we will have less ability to shape Britain's future than we have today. That is not what the public voted for."
Dr Fox refused to say whether he would quit if Mrs May changed course, adding: "Getting no answer you can draw your own inferences."
The Government warned giving MPs a decisive final say on the terms of the Brexit deal would weaken the Prime Minister's hand in negotiations.
A second defeat in a string of votes on Monday saw peers insist that Parliament should be given a say on the Government's mandate for trade talks with Brussels while a third was aimed at making sure refugees in Europe would continue to be allowed to join family members in the UK.
Peers backed an amendment to the European Union (Withdrawal) Bill by 335 to 244, majority 91, to give Parliament a decisive say on the outcome of the Brexit negotiations.
It would spell the end of the "take it or leave it" approach promised by ministers, under which the only alternative to accepting a deal thrashed out by Mrs May and Brussels would be the UK leaving without a deal.
Former deputy prime minister Lord Heseltine and ex-ministers Lord Patten of Barnes and Lord Willetts were among 19 Tory rebels to support the cross-party amendment.
-- LabourLordsUK (@LabourLordsUK) April 30, 2018
Shadow Brexit secretary Sir Keir Starmer urged the Prime Minister to accept the cross-party amendment, warning that there was "no majority in Parliament for a no-deal Brexit".
He said: "If Parliament votes down the Article 50 deal, then Parliament must decide what happens next.
"Under no circumstances can the Prime Minister be given a blank cheque to crash the UK out of the EU without a deal."
Senior Tory backbencher Sarah Wollaston said Brexiteers wanted to "take back control" to the UK but "Parliament won't support a hard Brexit".
If Labour want to be taken seriously as a party which respects the decision the people took in the EU referendum, they must now remove Lord Hunt from their front bench. https://t.co/MwOYf4mEeq
-- Steve Baker MP (@SteveBakerHW) April 30, 2018
Brexit Minister Steve Baker said: "If Labour want to be taken seriously as a party which respects the decision the people took in the EU referendum, they must now remove Lord Hunt from their front bench."