David Cameron evoked his own family's Scottish heritage as he delivered a heartfelt plea for Scotland to remain part of the United Kingdom.
In his highest-profile intervention in the debate on Scottish independence, the Prime Minister warned that the world would lose "something very powerful and precious" if the UK's "family of nations" broke up forever.
But Scotland's First Minister Alex Salmond accused Mr Cameron of making a "bogus" argument and repeated his challenge for the Prime Minister to agree to a head-to-head debate on the issue.
Mr Cameron called on the people of England, Wales and Northern Ireland to send a message to Scotland as it prepares to vote on September 18: "We want you to stay."
Independence would be bad for Scotland but would also leave the United Kingdom "deeply diminished" and would "rip the rug from under our own reputation" in the world, Mr Cameron said.
He warned supporters of the union that they have only "seven months to save the most extraordinary country in history".
Separation would not only cost the UK some of its economic, political and diplomatic "clout" in the world, but would also tear up an "intricate tapestry" of human connections and relationships which mean that "for millions of people, there is no contradiction in being proud of your Scottishness, Englishness and Britishness - sometimes all at once".
Recalling that the name Cameron stems from the West Highlands, the Prime Minister said he was "proud" of his Scottish heritage and pointed out that the clan motto is "let us unite".
Mr Cameron defended the decision to deliver the speech in London, arguing that he was making the case for the rest of the UK to speak out on the issue and stressed that he would soon be visiting Scotland along with the entire UK Cabinet.
The Prime Minister has previously acknowledged that his image as a "Tory toff from the Home Counties" does not make him a good figurehead for the No campaign in the referendum debate.
But he said: "F rankly, I care far too much to stay out of it. This is personal.
"Our great United Kingdom - brave, brilliant, buccaneering, generous, tolerant, proud - this is our country.
"And we built it together. Brick by brick, Scotland, England, Wales, Northern Ireland, brick by brick.
"This is our home - and I could not bear to see that home torn apart. I love this country.
"I love the United Kingdom and all it stands for. And I will fight with all I have to keep us together."
Speaking in the velodrome where he watched Scottish cyclist Sir Chris Hoy win Olympic gold in 2012, Mr Cameron said the London Games were an example of the "power of collaboration" which has given the nations of the UK a big place in the world.
Name-checking UK-wide assets from the BBC, the NHS and the armed forces to the country's place in the UN Security Council, Nato and the G8, and cultural "icons" such as Sherlock Holmes, Emeli Sande and Scotch whisky, the Prime Minister said: "We come as a brand - a powerful brand.
"Separating Scotland out of that brand would be like separating the waters of the River Tweed and the North Sea.
"If we lost Scotland, if the UK changed, we would rip the rug from under our own reputation. The plain fact is, we matter more in the world together."
He said there was a moral case for preserving the union which had made the UK "a country that has never been cowed by bullies and dictators, a country that stands for something".
Citing the inspiration that Britain had given to fighters for freedom such as Nelson Mandela and Aung San Suu Kyi, Mr Cameron said: " Our values are of value to the world. In the darkest times in human history there has been, in the North Sea, a light that never goes out.
"And if this family of nations broke up, something very powerful and precious would go out forever."
Mr Cameron acknowledged that many in England, Wales and Northern Ireland felt there was no place for them in the debate about Scottish independence, whether they were "quiet patriots" who thought there was little they could do to influence the outcome or "shoulder shruggers" who felt that separation would not matter much to life south of the border.
And he said there were "a few" who thought the rest of the UK would be better off without Scotland.
"All the above are wrong," said the Prime Minister. "We would be deeply diminished without Scotland."
But Mr Salmond accused the Prime Minister of being a member of an "out-of-touch Westminster elite" and insisted an independent Scotland would maintain close ties with the rest of the UK.
The SNP leader said: "This speech was a threadbare defence of the case for Westminster Tories retaining their undemocratic control over Scotland, which betrays the utter weakness of the Prime Minister's case.
"David Cameron said he will 'fight with all he has' against Scotland's independence - but that doesn't extend to having a head-to-head debate on the subject.
"The Prime Minister says his appeal today is to the people of England, Wales and Northern Ireland - but his Government makes policy for of all of the UK, including Scotland, and he cannot keep dodging that debate.
"He has the perfect opportunity later this month when our respective cabinets are meeting in the Aberdeen area just a few miles apart on the same day, and I challenge him again to have that debate and let the people hear the arguments on each side."
Rejecting the emotional case for the union set out by Mr Cameron, the First Minister said: "An independent Scotland will be the best of friends and neighbours with the other nations of these islands, with the closest of family, cultural and business ties.
"The people of England would rather he turned his attentions to issues like the flooding in Somerset or the policies which see a growing wealth and equality gap between London and the rest of England, which an independent Scotland will help counterbalance."