Sir Vince Cable has become the new leader of the Liberal Democrats with a promise to offer voters the possibility of an "exit from Brexit" through a second referendum on EU membership.

The former business secretary took up the role after nominations for the post closed without him facing any challengers, becoming, at 74, the oldest leader in the party's history .

He said that he aimed to ensure that the Lib Dems occupy the "gigantic space in the middle of British politics" left by the two main parties' move to the extremes.

The vacancy for the leadership came after Tim Farron announced he would stand down because he had been unable to reconcile his Christian faith with the demands of leading a "progressive, liberal" party.

A disappointing general election campaign saw the party increase its tally of MPs by just three to stand at 12.

Speaking in Westminster shortly after being named leader, Sir Vince said the Government's approach to Brexit was taking Britain towards the "disastrous outcome" of crashing out of the European Union.

Negotiations were being conducted by a "dysfunctional, disorganised, disunited" administration, whose strategy was drawn up before the full complexity of Brexit was apparent and at a time when Theresa May had "serious political authority", which she has now lost, he said.

Liberal Democrats would work with like-minded people in other parties to fight to keep UK participation in the single market and customs union and continued collaboration on research with European partners, he said.

He said: "What we now need is an exit from Brexit.

"The exit from Brexit comes as a result of the policy that we have adopted, which is that we must consult the British public at the end of the process."

Voters should be asked "Do you wish to accept what is coming down the track, jumping off the cliff and hoping there's a tree to catch you or do we want to stay within the European Union?", he said.

Following last year's referendum, Sir Vince set his face against Mr Farron's offer of a second referendum, describing it as "seriously disrespectful and politically utterly counterproductive".

But he threw his weight behind the idea earlier this week, insisting a new referendum would not be a re-run of the 2016 vote, but a new question in changed circumstances.

Sir Vince told activists: "We have gigantic space in the middle of British politics.

"The two major parties have been captured by ideologues, ideologues on the one hand who hate Europe and on the other hand who hate capitalism.

"As a result, British politics is more polarised and more divided than at any time any of us can remember.

"What is now badly missing is the basic commonsense and moderation and mutual respect that are what British politics is at its best.

"My aim is that our party and I will occupy that space in British politics."

Spelling out his political philosophy, Sir Vince added: "In my view the world isn't just about economics and making numbers add up, though that's important.

"I came into politics as a radical and a reformer and I want to put at the centre of what I do addressing some of the inequalities that disfigure British society.

"I think that can be done.

"I think it can be done because at heart the British public are humane and tolerant and I think we can appeal to that instinct, which is in very marked contrast to the icy indifference of this Conservative government."

He said he was "ambitious for this country and ambitious for our party".

"In difficult times we have shown enormous resilience but I now believe that we can fight our way back, break through and make an enormous success of our party and, eventually, in government."

Sir Vince insisted he was not a stop-gap leader, saying: "I will serve for as long as I need to. I'm not here for the short term, I'm not here as a caretaker."

The Lib Dem leader said Chancellor Philip Hammond deserved some opposition support against more hardline Brexiteers in Cabinet.

"He is a political adult, you've got Boris Johnson in short trousers, and Dr Fox in nappies. In that company he deserves some qualified support."

Sir Vince drew comparisons between the post-referendum atmosphere and the one after Enoch Powell's anti-immigration "Rivers of Blood" speech in 1968.

"There is very little doubt that the emotions aroused by the Brexit debate, although they are not the same, have a very high element of xenophobia, and hostility to foreigners in general.

"And there are people who were at the heart of that debate who deliberately sought to whip up that feeling.

"So, although they are not the same, they have very much diminished the quality of political debate."

Sir Vince told BBC Radio 4's PM that the Liberal Democrats' anti-Brexit position at last month's election was "fundamentally right but premature", and will be "absolutely vindicated as right" as the realities of EU withdrawal become clear.

The new leader's age has led to speculation at Westminster that Sir Vince will effectively be a caretaker leader, handing over to deputy Jo Swinson nearer the time of the next general election.

But Ms Swinson told PM: "There's been no kind of discussion like that. We will obviously work very closely together as we have before. He was Business Secretary and I was, first of all, his aide, and then I was a minister in that department and we had a really constructive working relationship."

Asked why she did not stand for the top job herself, as some had expected, the 37-year-old East Dunbartonshire MP said: "I thought about it and came to the conclusion this wasn't the right thing to do and deputy leader was the way in which I can contribute to the Liberal Democrats."

Former Lib Dem leader and deputy prime minister Nick Clegg said: "Congratulations and good luck to Vince as new leader of the Liberal Democrats. As Brexit engulfs the UK economy, his experience and insight is needed more than ever."