In an attempt to stop the mass resignation of Cabinets Remainers, Mrs May last month offered the House the opportunity to vote to take a no-deal off the table should she lose her second vote, and if a clean break is voted against, allow MPs to vote to extend Article 50, thereby delaying the country's exit from the EU.
"We regret the outcome of tonight's vote", the spokesperson said.
The "public vote" part apparently refers to a new referendum on Brexit, as Labour has seemingly walked away from its support towards putting May's deal on public vote that it expressed just a week ago.
The motion was deliberately vague, essentially saying that the government does not wish to actively pursue no-deal Brexit as official policy, yet acknowledged that no deal is the default option, should nothing be agreed before March 29.
This is because there are a handful of non-voting members of the house such as the Speaker John Bercow and Sinn Fein members who do not take up their seats. "In the light of our own legal analysis and others, we do not recommend accepting the government's motion today", said Tory MP Bill Cash, who is an ERG member.
'I think it is still alive, I do, ' Ms Truss told BBC Radio 4's PM.
Extending the timeframe for Brexit requires approval from all 27 remaining European Union member countries.
A third vote Wednesday further affirmed Parliament's conviction to reject a no-deal Brexit under any circumstances - it passed 321 for and 278 against.
Although Britain is now looking to leave the European Union only after having secured a deal with Brussels, it still needs to negotiate a delay on Article 50 by March 29 - the Brexit deadline day.
If no deal is agreed by March 20, "then it is highly likely the European Council at its meeting the following day would require a clear goal for any extension, not least to determine its length, and any extension beyond 30 June 2019 would require the United Kingdom to hold European Parliament elections in May 2019", Thursday's motion says.
"It is disappointing that the United Kingdom government has been unable to ensure a majority for the withdrawal agreement agreed by both parties in November, even after multiple additional assurances from the EU".
Some British lawmakers had warned their Brexit-backing colleagues ahead of Tuesday's vote that rejecting the deal could lead to Britain's departure being postponed indefinitely, because a delay would give momentum to opponents of Brexit.
As one of our loyal readers, we ask you to be our partner.
The "no deal" vote emphasised May's lack of control over events.
Many fear Brexit will divide the West as it grapples with both the unconventional U.S. presidency of Donald Trump and growing assertiveness from Russian Federation and China, leaving Britain economically weaker and with its security capabilities depleted.
Both lawmakers and the public remain split between backers of a clean break from the European Union and those who favor continuing a close relationship, either through a post-Brexit trade deal or by reversing the June 2016 decision to leave.
The EU's chief negotiator Michel Barnier told the European Parliament it was the responsibility of London to suggest a way forward. Juncker on Monday said Brexit "should be complete before the European elections" at the end of May.
"Not for us, not for Britain and certainly not for our citizens", he said. That extension could be as short as three months or as long as two years.
After MPs crushed her deal for a second time on Tuesday, May said it was still the best option for leaving in an orderly fashion.