David Cameron has said that Scotland could receive more powers if its people vote to remain part of the UK.
The Prime Minister was speaking after talks with Scottish First Minister Alex Salmond over a referendum on independence.
Speaking shortly after the meeting in Edinburgh, Mr Cameron said his "absolute priority" was to preserve the United Kingdom.
Mr Cameron said: "If the answer to the question is that Scotland wants to stay in the United Kingdom, then further options for devolution are on the table.
"We can discuss those just as we have in the past, to make sure that all the parts of the United Kingdom feel that they get a good deal inside the United Kingdom."
Mr Cameron described his discussions with Mr Salmond as "constructive", but said: "On the issue of independence, separating Scotland, leaving the United Kingdom, I am afraid there wasn't much progress.
"I believe that we need to put a straightforward and simple and legal question to the Scottish people in good time, which is to ask the straightforward question 'Do you want to stay in the United Kingdom...' - and I hope that is what people will vote for - '... or do you want to leave the United Kingdom?"'
"My absolute priority is to keep the United Kingdom together.
"It has worked for all our countries, it's made us safer, it's made us stronger, it's made us richer, it's helped us in times of difficulty. We shouldn't give up this great thing - our United Kingdom."
Earlier, the Prime Minister used a speech in Edinburgh to set out his defence of more than 300 years of political union between Scotland and England.
The referendum result would not be "the end of the road", said Mr Cameron.
"When the referendum on independence is over, I am open to looking at how the devolved settlement can be improved further.
"And, yes, that does mean considering what further powers could be devolved.
"But that must be a question for after the referendum, when Scotland has made its choice about the fundamental question of independence or for the United Kingdom, when Scotland has settled this question once and for all - and ended the uncertainty that could damage and hold back Scotland's prospects and potential."
In his keynote speech, Mr Cameron said he was "ready to fight for the life of this country".
Although he acknowledged that Scotland - and England - could make their ways as independent states, he said he was "convinced" the UK's best days lay ahead.
Mr Cameron said: "I come here today with one simple message: I hope and wish that Scotland will vote to remain part of the United Kingdom."
Describing the UK as more than "some sort of deal", he added: "It's a precious thing. It's about our history, our values, our shared identity and our joint place in the world.
"From Waterloo to the Second World War, our servicemen and women have fought and won together. Your heroes are our heroes."
On the international stage, he said the UK is stronger together with a place in Nato, Europe and the UN Security Council.
Economically, he said, Scotland's population of about five million is stronger in the larger UK.
He said the United Kingdom stands for common endeavour, concluding: "If anything's worth fighting for, that surely is - which is why I'm ready to fight for the life of this country."
Mr Cameron said his part in the constitutional debate was not motivated by party interest, acknowledging the sole Tory MP north of the border.
"I know the Conservative Party isn't currently - how can I put this? - Scotland's most influential political movement," he said.
"So, more than a little humility is called for when any contemporary Tory speaks in Scotland. In fact, some say it might be wiser not to speak at all."
Mr Cameron conceded that pro-union politicians needed to do more get their message out to voters and suggested he hoped Labour's Alistair Darling and Gordon Brown would get involved.
Pressed on whether his privileged background would deter Scottish voters from backing the pro-union campaign, he said: "As for the fact that I am English, that I had a privileged upbringing, that I might annoy people by making a speech in Scotland, all I can say is that this is what I believe. I care about our United Kingdom."
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