In his most high-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.
Speaking at the Olympic Park in east London, 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.
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: "I am as proud of my Scottish heritage as I am of my English heritage.
"The name Cameron might mean 'crooked nose' but the clan motto is `Let us unite' - and that's exactly what we in these islands have done."
Mr Cameron was accused by Scottish National Party Deputy First Minister Nicola Sturgeon of a "shameful" attempt to hijack the Olympic spirit for political purposes on the day when the Winter Games begin in Russia.
And Ms Sturgeon branded him "cowardly" for refusing to debate head to head with First Minister Alex Salmond.
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: "Frankly, 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."
Warning supporters of the Union that they have only "seven months to save the most extraordinary country in history", he said: "Think of what we've done together - what we can do together - what we stand for together.
"Team GB. The winning team in world history. Let us stick together for a winning future too."
The Prime Minister said it would be "extremely difficult to make a currency union work between an independent, separate Scotland and the rest of the UK".
He added: "We would be diminished if Scotland were to leave our extraordinary, successful and remarkable family."
At a venue chosen to symbolise the successes of the whole United Kingdom working together as Team GB, Mr Cameron said that the Olympic medals were won under the banner of a union flag that was not only red and white but also blue.
The London Games - when 14 of Team GB's 65 medals were won by Scots including Andy Murray and Sir Chris Hoy - were an example of the "power of collaboration" which has given the nations of the UK a big place in the world, he said.
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 like 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."
And he said that 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... Our shared values - Freedom. Solidarity. Compassion. Not just overseas, but at home."
Citing the inspiration which British values gave to fighters for freedom like Nelson Mandela and Aung San Suu Kyi, Mr Cameron said: "Our Parliament, our laws, our way of life so often, down the centuries, the UK has given people hope.
"We've shown that democracy and prosperity can go hand in hand that resolution is found not through the bullet, but the ballot box.
"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.
"This matters to all our futures. And everyone in the UK can have a voice in this debate."
Responding to Mr Cameron's speech, British Chambers of Commerce director general John Longworth said: "While an independent Scotland could be a viable country, the break-up of the Union between Scotland and the rest of the UK would certainly be disruptive to business, to the economy, and to British politics as we know it.
"Scotland leaving the Union would be a much bigger deal even than the UK leaving the EU - as the constitutional and economic ties are even greater. Crucially, in both cases, there would be no turning back."
The chief executive of manufacturers organisation EEF, Terry Scuoler - a Glaswegian and former captain in the Argyll and Sutherland Highlanders - said: "The Prime Minister is right to set out the many positive economic and political reasons why Scotland should remain part of the United Kingdom.
"I personally understand the emotional drivers behind the idea of an independent Scotland. However, I think the longstanding economic and cultural ties with the UK represent a far stronger gravitational force in favour of the union.
"Many businesses are now setting out their concerns, putting their heads above the parapet and saying why they are opposed to an independent Scotland. Many others are telling me privately that they are against any suggestion of Scotland ejecting itself from the UK. The enemy of investment and stability in business is uncertainty. While the economic risks around independence are not fully quantified, they do represent a significant gamble."
Mr Salmond said Mr Cameron had presented a "very flimsy case" in a speech "largely devoid of meaningful content".
He told BBC Radio 4's World at One: "The reason for the change of approach is the last seven opinion polls that show the Yes campaign gaining ground. Still behind in the opinion polls, but each one of them showing the Yes vote increasing.
"So I think there's a bit of panic in the breastie of the Prime Minister."
He mocked the Prime Minister's exhortation to people in the rest of the UK to phone Scottish friends and urge them to vote No.
"If I were in the Somerset Levels just now, that wouldn't be my priority. My priority would be to lift the phone to Downing Street to tell the Prime Minister to get his wellies on and come and help them, as opposed to being asleep on the job which he's been since December," the First Minister said.
"I don't think this is a prime minister who speaks for Britain or, for that matter, for England. I think he speaks for a Westminster elite with Westminster elite priorities. That is Cameron's weakness."