Theresa May promised to build a "better Britain" and to make the UK's EU exit a "success" after she was announced as the new Tory leader and soon-to-be PM.
Speaking outside Parliament, Mrs May said she was "honoured and humbled" to succeed David Cameron, after her only rival in the race withdrew on Monday.
Mr Cameron will tender his resignation to the Queen after PMQs on Wednesday.
Mr Cameron, who has been UK prime minister since 2010, decided to quit after the UK's Brexit vote.
It follows another day of dramatic developments in the political world, when Andrea Leadsom unexpectedly quit the two-way Conservative leadership contest, saying she did not have the support to build "a strong and stable government".
Her decision left Mrs May - the front runner - as the only candidate to take over leading the party and to therefore become prime minister.
In a speech flanked by dozens of Conservative MPs, Mrs May, the home secretary since 2010, praised Mr Cameron for his stewardship of the Tory party and the country.
And she paid tribute to Mrs Leadsom for her "dignity" in withdrawing her leadership bid, as well as to the three other candidates who ran in the contest.
"I am honoured and humbled to have been chosen by the Conservative Party to become its leader," Mrs May told the gathered media.
She said her leadership bid had been based on the need for "strong, proven leadership", the ability to unite both party and country and a "positive vision" for Britain's future.
"A vision of a country that works not for the privileged few but that works for every one of us because we're going to give people more control over their lives and that's how, together, we will build a better Britain."
And in a message perhaps designed to reassure Brexit-supporting colleagues, Mrs May - who campaigned to stay in the EU, said: "Brexit means Brexit and we're going to make a success of it."
What happens next?
- The 1922 Committee of Conservative MPs, which is overseeing the leadership contest, has declared Mrs May the new party leader "with immediate effect".
- David Cameron says he will take Prime Minister's Questions on Wednesday and then head to Buckingham Palace and officially tender his resignation to the Queen and recommend she sends for Theresa May as his replacement
- Mrs May will then go to Buckingham Palace to see the Queen and receive her invitation to form a government
- Theresa May should then be in place as UK prime minister by Wednesday evening - it is not yet clear when the Cameron family will move out of No 10
Earlier, in a brief statement outside No 10, Mr Cameron said he was "delighted" that Mrs May was to succeed him in Downing Street.
He said a "prolonged period of transition" was not necessary, and added: "So tomorrow I will chair my last cabinet meeting. On Wednesday I will attend the House of Commons for Prime Minister's Questions.
"After that I expect to go the Palace and offer my resignation."
The prime minister praised Mrs May as "strong" and "competent" and he said she was "more than able to provide the leadership" the UK needs in the coming years.
"She will have my full support," he added.
Key dates for the new PM
- 18 July - Parliament due to vote on Trident renewal
- 19 July - Possible date for her first cabinet meeting
- 20 July - First PMQs as prime minister
- 5 September - Parliament returns from summer recess
- 2-5 October - Conservative Party annual conference
- 20 October - Her first European Council meeting as prime minister
Announcing her decision to pull out of the contest, Mrs Leadsom - who was a leading light of the Brexit campaign - said a nine-week leadership campaign at such a "critical time" for the UK would be "highly undesirable" - and she gave her backing to Mrs May.
A source close to the energy minister told BBC political editor Laura Kuenssberg "the abuse has been too great" for Mrs Leadsom during the contest.
Mrs Leadsom had apologised to Mrs May on Monday after suggesting in a weekend newspaper interview that being a mother made her a better candidate for the job.
BBC chief political correspondent Vicki Young said Mrs May had begun the day launching her leadership campaign to take to the party membership - and within the space of several hours found out she would be prime minister by Wednesday.
Mrs May would now have to decide the make-up of her new cabinet, she said.
Mr Cameron announced his intention to resign as prime minister on 24 June, after finding himself on the losing side of the EU referendum, with the UK voting by 52% to 48% in favour of leaving.