Prime Minister Theresa May has called on her fellow Conservatives to continue "loudly and clearly" putting the case for the "precious Union" of the UK.
During a speech to the Scottish Tory conference in Glasgow she said "we are four nations, but at heart one people".
Mrs May had earlier criticised the SNP-led government at Holyrood for having tunnel vision over a second Scottish independence referendum.
The PM insisted that "logic and facts" were on the side of the UK.
Scotland's First Minister Nicola Sturgeon has maintained that a second referendum was "highly likely" because of Brexit.
Voters in Scotland backed remaining in the EU by 62% to 38%. The UK as a whole voted to leave by 52% to 48%.
On the issue of Brexit, Mrs May told the conference that she was "determined" to ensure that "we leave the EU as one United Kingdom and prosper outside the EU as one United Kingdom".
'Feeble and incompetent'
She began her speech with a scathing attack on Labour in Scotland.
Mrs May said: "For too long a feeble and incompetent Scottish Labour opposition did nothing to scrutinise the SNP for their failures.
"An SNP government interested only in stoking-up endless constitutional grievance and furthering their obsession with independence, at the expense of Scottish public services like the NHS and education, was given a free pass by Labour."
The address had a strong pro-Union theme.
Mrs May said the the UK "we cherish" is not a thing of the past.
She added: "The Union I am determined to strengthen and sustain is one that works for working people across England, Scotland, Wales and Northern Ireland."
Analysis - By BBC Scotland political editor Brian Taylor
One or two of Mrs May's predecessors in Downing Street have had a bash at defining Britishness or British values. Not with universal success. One thinks of cycling and evensong.
But Theresa May's attempt appeared to be more broadly grounded and thus more interesting. She spoke again of family, of shared endeavour, of common interests.
Admittedly, some of her examples were less than utterly convincing. She talked of the Harry Potter books being written in an Edinburgh cafe by a Gloucestershire author. Thus, apparently, stressing the benefits of Union.
Well, maybe. But George Frederick Handel composed many of his masterpieces in London. That did not mean that Britain and his native Germany were bound by musical or other ties.
Still, it was a thoughtful speech, giving the impression of much more than a stock oration culled from party archives.
Analysis - By The BBC's Scotland editor Sarah Smith
So will there be another Scottish independence referendum? Very possibly.
With all the talk of another vote being "very likely" and "all but inevitable", Nicola Sturgeon has already marched her troops so far up the hill it's hard to see how she can march them back down again without massive loss of face and political capital.
Today we got an extensive preview of the unionist case.
Mrs May talked at length about the benefits being part of the United Kingdom has bestowed on Scotland and the rest of the UK. And warned of the economic dangers of independence.
It was a serious and substantial speech about why she believes the UK is "Better Together".
Other things the speech touched on included:
- The "falling standards" of education in Scotland and the continued attainment gap. Mrs May said fewer young people from the poorest backgrounds made it to university in Scotland compared with the rest of the UK
- Praising Scottish Conservative leader Ruth Davidson whom she said would stand up to the SNP establishment
- "Enduring" economic strengths of the UK linked to its integrated domestic market for businesses meaning "no barriers to trade within our borders"
- The importance of the national security of the Union in a "changing world". Mrs May said: "The United Kingdom is a responsible member of the international community and Scotland makes a huge contribution to the UK's global role."