David Cameron was speaking as draft clauses that will form the basis of new laws were published by the UK Government.
The Prime Minister, who is visiting Scotland today, together with the other main Westminster leaders had pledged to deliver new powers to Holyrood if Scots rejected independence in last year’s referendum.
A total of 44 draft clauses, which will underpin new legislation, have now be published.
Mr Cameron has already pledged they will become law in the next parliament, regardless of who wins May’s general election.
UK Government ministers insisted the proposals will transform the Scottish Parliament into one of the most powerful devolved assemblies anywhere in the world.
The Scottish Government has welcomed the changes – which had been put forward by the Smith Commission – but has argued that the new powers do not go far enough.
Mr Cameron said: “In September the people of Scotland came out in record numbers to decide the future of the United Kingdom.
“They voted clearly and decisively to keep our family of nations together. But a No vote did not mean no change.
“The leaders of the other main political parties and I promised extensive new powers for the Scottish Parliament – a vow – with a clear process and timetable.
“And now here we have it: new powers for Scotland, built to last, securing our united future.”
The legislation will state clearly for the first time that the Scottish Parliament is a permanent institution.
Powers over Holyrood and local government elections north of the border will also be handed to MSPs.
New tax powers will see Scotland get the ability to control and set income tax rates and bands, and will also mean cash raised from the levy north of the border will stay in Scotland, rather than go to the Treasury.
As a result, the Scottish Government’s block grant from Westminster “will be reduced to reflect the tax revenues that the UK Government will forego as a result”, the command paper sets out.
Powers over air passenger duty (APD) are also to be devolved, with the clauses including “provision for appointing the day when APD will be switched off in relation to Scotland”.
Holyrood will also get new powers over welfare, and although the new Universal Credit will remain under Westminster control, the Scottish Parliament is to be given the power to vary the housing cost elements of this, including the controversial under-occupancy charge – branded the “bedroom tax” by its opponents.
“This means that Scottish ministers will be able to decide whether to apply any under-occupancy reductions,” the command paper states.
Scottish First Minister Nicola Sturgeon claimed there had been a “significant watering down” of key Smith proposals on welfare, employment support and borrowing.
The welfare provisions will not allow the Scottish Parliament to create new benefit entitlements in devolved areas, the Scottish Government said, and would also mean UK ministers would need to give their approval to any changes to the Universal Credit, including the so-called “bedroom tax”.
Ms Sturgeon said: “T hroughout this process, I have been clear that, despite it falling short of the real home rule powers we need to create jobs and tackle inequality, the Scottish Government would be a constructive participant, working with the UK Government to bring forward what Lord Smith recommended.
“The legislation published today does not represent the views of the Scottish Government, but it does represent some progress.”
But she added: “Too much of what the Prime Minister has set out imposes restrictions on the recommended devolved powers and would hand a veto to UK ministers in key areas.
“For example, the proposals on welfare do not allow us to vary Universal Credit without the permission of the UK Government. That means – under the current proposals – we will not have the independence to take action to abolish the bedroom tax.
“At the same time, the power argued for by stakeholders to create new benefit entitlements in any devolved area has simply not been delivered, while the command paper makes clear that, pending devolution of disability support, the roll-out of personal independence payments and the cut to spending on disability benefits will continue.
“This cannot, under any interpretation, represent the meaningful progress on the devolution of the powers we need to design a social security system that meets Scotland’s needs.”
Ms Sturgeon continued: ” The support for unemployed people also falls short of what Lord Smith recommended, with the provisions set out today narrowly focused on existing schemes.
“And the paper confirms that the Scottish Government will still have to work within the framework of austerity being imposed by the UK Government. It also suggests that Scotland’s capital grant could be replaced by borrowing powers and not augmented by them as was clearly the intention of the Smith proposals.
“In these crucial areas the clauses set out today appear to be a significant watering down of what was promised by the Smith Commission and need an urgent rethink by the UK Government.”
The First Minister said her administration “r emain committed to this process, despite the difficulties we have experienced in getting information in a timely fashion”.
She added: “We will continue to work with the UK Government and other stakeholders to ensure that the changes are made ahead of the Bill being taken through Westminster.
“Ultimately, however, the decision on whether the Smith proposals go far enough in delivering the powers we need to create prosperity, tackle inequality and protect our public services will be for the people of Scotland to take.”
Scottish Secretary Alistair Carmichael insisted Ms Sturgeon “is wrong” to suggest UK ministers retain a veto in key areas.
He said anyone who attempts to use the Smith powers to “thwart” the effective working of the two governments “demonstrates a fundamental lack of respect for the democratically expressed wishes of the Scottish people from the referendum.”
“There is no veto,” he said at a briefing at Edinburgh’s Dynamic Earth science museum.
“It’s a mature conversation between two governments. Federal governments across the world manage it all the time. I’m sure it’s within our range of responsibilities.
“If you are to accept the (veto) analysis then you would have to accept that a veto would pull in the other direction as well and I just don’t see that as sensible or workable.
“It’s not what is intended and it’s disappointing but not surprising that it is being suggested.”
The Government command paper said one of the next steps in implementing the Smith proposals will be to “provide for effective and workable mechanisms to resolve inter-administration disputes … with a provision for well-function arbitration processes as a last resort”.
Mr Carmichael added: “That’s a discussion that we now have to have between ourselves and the Scottish Government.
“It is going to be more important than ever that Scotland’s two governments are able to work together in a mature, co-operative and collaborative way if this settlement is going to work – and it has to be able to work because that is what Scotland told us they wanted to do on September 18.
“Anybody who approaches the exercise of these powers with a view to thwarting the effective working of the two governments demonstrates a fundamental lack of respect for the democratically expressed wishes of the Scottish people from the referendum.”
Arbitration is expected to take place in the existing Joint Ministerial Committee of the UK Government and devolved administrations, a Scotland Office official said.
Mr Carmichael added: “It would be refreshing if, instead of trying to kick up dust like this, the nationalists would now tell us what they want to do with the powers of the third most powerful devolved parliament in the world. That is where I think the debate in parliament needs to go.”
The SNP has suggested it may push for “full fiscal autonomy” at the general election.
Mr Carmichael said an SNP majority in Scotland “could give them a mandate to demand” full fiscal autonomy but said the present command paper is “a prospective that can be delivered across the whole of the UK and it is one that the SNP signed up to”.
“What they call full fiscal autonomy is in fact independence through the back door, and frankly that demonstrates a total lack of respect, I think, for the vote on September 18,” he said.
Chief Secretary to the Treasury Danny Alexander dismissed Ms Sturgeon’s suggestion that the proposals do not allow the Scottish Government to vary Universal Credit without the permission of the UK Government or abolish the “bedroom tax”.
“The Scottish Government has the power to make variations to housing benefit charges in whatever way it wanted to,” he said at the Edinburgh briefing.
“The consultation exists because there are UK responsibilities and Scottish responsibilities and the two governments need to talk to one another.
“There is no veto implied in this – it is just a matter of the normal functioning of government.
“In a sense this is a way, in legislation, that we make that clear.
“This does not give a veto, it just requires mature consultation which is frankly what already happens in a lot of areas.”
He added: “These clauses give Scottish ministers the power to deliver the changes in the areas that are set out in the Smith Commission – no ifs, and no buts.”
Downing Street said that, in face-to-face talks with Ms Sturgeon, Mr Cameron would challenge the SNP leader over her suggestion that the party’s MPs could vote on matters relating to the NHS in England in the event of another hung parliament after the general election in May.
Ms Sturgeon said the move was necessary as health spending in Scotland was linked to spending levels in England, but Conservative defence minister accused her of putting the Union at risk.
A Number 10 spokeswoman said: “As I understand it the SNP by convention has never voted on English-only matters.
“(The Prime Minister’s) view is the principle that decisions affecting only England should be decided by a majority of English MPs is a fair principle.
“She has obviously raised this in advance. I expect it will be discussed today.”
Lord Smith, who chaired the cross-party commission on more powers for Scotland, said publishing the draft clauses was “an important step forward”.
He said: ” In November, five parties came together to reach an unprecedented agreement on new powers for the Scottish Parliament. Today marks an important step forward in the enactment of the agreement and it is vital that both governments work together closely and constructively to see the job through.”
Deputy Prime Minister Nick Clegg said: ” Today is a proud day for the people of Scotland and for Liberals, who have campaigned for Home Rule for more than a century.”
The Liberal Democrat leader said: ” We have delivered on Home Rule for Scotland within a strong United Kingdom – and that is what people in Scotland said they wanted.
“We passionately campaigned for Scotland to stay part of our family of nations because we firmly believe we are better together.
“During that referendum campaign I signed a vow with other UK party leaders promising extensive new powers if the people of Scotland voted to stay in the United Kingdom. The people of Scotland decisively rejected independence and we have honoured the vow.
“Liberal Democrats in Government have not only met the timetable set for more powers for Scotland but we have completed it ahead of schedule.
“The Scottish Parliament will raise over half of the money it spends, we will have a Scottish welfare system with a starting budget of over £2.5 billion and there will be votes for 16 and 17-year-olds for Holyrood and local government elections.”