Barnier has long made clear, however, that May's rejection of remaining in the customs union or single market would mean administrative hurdles to the movement of goods.
On the matter of European Court of Justice jurisdiction, she noted that its existence, and its rulings, will still impact the UK.
But she also says Britain is committed to avoiding a hard Irish border and ensuring there is no return to violence in Northern Ireland.
"People voted to take back control of our money, our laws and our borders and that's exactly what we will be doing".
May called the EU's proposal unacceptable.
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Following is a summary of the main points made by May in her speech.
May is making a speech outlining her vision of future economic ties with the EU.
Lastly, May vowed to keep United Kingdom regulatory standards "as high as the EU's" to ensure further trade can be as smooth as possible.
TechUK CEO Julian David said the tech industry welcomed May's "crystal clear commitment" on data protection, and that the move to go further than the adequacy arrangement "is the right approach to support the UK's place as a leading tech economy".
And a "customs partnership" in which the European Union trusted Britain to collect tariffs for it was "fantasy", he added - the European Union is already suing over customs fraud in Britain and long memories of Britain's "mad cow" scandal two decades ago have left Brussels wary of accepting British standards in, say, animal health.
The EU has already dismissed this as "cherry-picking", but May pointed out that each of the bloc's existing trade deals with other countries was slightly different.
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The Prime Minister also addresses the issue of Northern Ireland, on which there has been increasing focus, with Labour pegging its decision to push for a customs union with the EU on the need to resolve the issue in a way that maintained the hard fought for peace process. In fact, arch-Leave supporter Iain Duncan Smith has already welcomed the speech, calling on the European Union chiefs to "stop playing games and stop the megaphone and start negotiating as equal partners".
She touched on many specific areas, explaining that the United Kingdom will not be pursuing passporting rights for financial services, but would want to achieve associate membership of European Medicines Agency, European Aviation Agency, and other agencies.
"It would also require us to be subject to a single rule book, over which we would have no say", said May. EPC Chief Executive Fabian Zuleeg, who moderated the debate, didn't believe in the possibility: "The only alternative to hard Brexit is no Brexit, but clearly time is running out now".
"Our departure poses particular challenges for Northern Ireland and for Ireland".
The British and EU teams are set to return to the negotiating table for the entirety of next week, to hammer out the details of the Brexit transition period before this month's European Council meeting.
Prime Minister lays out the "hard facts" on Brexit, hits back at critics in Brussels and tells others that there is a need for compromise.
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Ms Sturgeon said: "She accepted that access to the single market - the world's biggest marketplace and one around eight times bigger than the UK's alone - would be reduced, yet said she would leave the single market and customs union anyway, even though her own government's analysis shows this will cost jobs and cut living standards".
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