Mike Russell, the Constitutional Relations Secretary in Nicola Sturgeon’s cabinet, said that people “require” such a ballot.
The comments appear to be among the strongest yet from an SNP politician in favour of a so-called People’s Vote on the terms of Brexit.
Scottish Government ministers have already said they would be “sympathetic” to this, but have been wary in case a second European referendum sets a precedent for what would happen if Scots were ever to vote for independence.
Mr Russell stressed however that there would need to be clarity on what would happen if Scotland again voted against Brexit and the UK voted for it.
The Constitutional Relations Secretary spoke out as Theresa May gathered her UK Cabinet at Chequers to discuss what Britain’s relationship with the EU could look like after the Brexit process.
In terms of getting a workable proposal that would be acceptable to the other 27 EU nations, Mr Russell said the Prime Minister was “very far from being out of the wood”.
He told BBC Radio Scotland’s Good Morning Scotland programme that this was the “issue that is going to haunt them”.
Pressed on whether there should be a second European referendum, he stated: “I’ve been very sympathetic to that, the only caveat I’ve expressed, and it needs to be resolved, is what happens if that vote were to repeat what happened two years ago - Scotland to vote to stay and the rest of the UK to vote to leave.
“We couldn’t do that twice. So that has to be resolved.
“But I think people do require a say, and we certainly require a say on the final outcome. This is not a second vote, this is a say on the final outcome.”
Mr Russell continued: “We are in the most extraordinary, the most difficult times that I can certainly remember. They are not being helped by certainly the most incompetent UK government I have ever seen.
“So there needs to be a clear choice. If the choice is Brexit Britain in this chaos and mess or independent membership of the EU, that would be a real choice.”
The Liberal Democrats, who have been campaigning for a second Brexit vote, welcomed Mr Russell’s remarks.
Scottish Lib Dem Europe spokesman Tavish Scott said: “It appears the SNP are finally starting to see the light. Mike Russell is right that the British people deserve to have the final say on whether Theresa May’s ropey Brexit deal is good enough.
“He does, however, need to give up on the constant caveats. This isn’t about political posturing. It’s about warding off economic disaster.
“We’re two years into this process and the Cabinet is in the midst of yet another crisis meeting to settle on its position. That is absolutely ridiculous.
“Political momentum is building for a People’s Vote and hundreds of thousands have taken to the streets to fight for one. All pro-Europeans need to get on side before it is too late.”