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8 year oldMrs Leadsom said she did not believe she had sufficient support to lead a "strong and stable government".
She also said a nine-week leadership campaign at such a "critical time" for the UK would be "highly undesirable".
The energy minister said Mrs May was "ideally placed" to implement Brexit, and wished her the "greatest success".
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 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 assistant political editor Norman Smith said Mr Cameron's successor could now be in place "much earlier than 9 September" - which is when the contest was due to finish.
It will be up to the 1922 committee of Conservative MPs to decide on a revised timetable or whether Mrs May should become leader, uncontested.
If it was the latter, there would then have to be discussions about when she took over as prime minister from David Cameron.
The time between Gordon Brown winning the Labour leadership uncontested and succeeding Tony Blair as prime minister was 38 days.
Mrs Leadsom - who was a leading light of the Brexit campaign - made it in to the final two, alongside Mrs May - who campaigned for Remain - last week.
She secured the support of 84 MPs - including former Tory leader Iain Duncan Smith and Boris Johnson - compared with Mrs May's 199 votes. Justice Secretary Michael Gove was eliminated after coming third.
There had originally been five contenders to succeed Mr Cameron, with MPs voting in two rounds to get that number down to two - with party's 150,000-strong membership to have the final say.
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