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.
The BBC reports 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."
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.