Mrs May got 165 of the 329 votes cast by Tory MPs.
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The minister, who voted against same-sex marriage, said claims on social media that he supported so-called “gay cure therapies” were fabricated.
Michael Gove was the Education Secretary between 2010 and 2014, and now serves as the Ministry of Justice.
Five candidates stood in the ballot, which saw disgraced former cabinet minister Liam Fox finish last and exit the race. The victor is due to be declared by September 9.
A poll for the ConservativeHome website put support among members for May on 37 percent, with 38 percent backing Leadsom, who was also prominent figure in the Brexit campaign.
Andrea Leadsom on the campaign trail.
Theresa May has insisted she is the only contender to succeed David Cameron as prime minister who can unite the country and secure the best Brexit deal after emerging as the frontrunner in the race.
Theresa May after winning the first round of the Conservative Party leadership election.
The next round of voting will then take place on Thursday and the process will continue until just two candidates remain.
Mrs Leadsom also has the support of two Remain MPs.
The victor of that vote will become the new Conservative leader and prime minister of the United Kingdom.
Fifth place finisher Liam Fox has been eliminated from the next round of voting.
Leadsom has defined herself in her pitch to Tory activists as a candidate who can be trusted to take the United Kingdom out of the European Union, unlike Theresa May who campaigned for Remain during the referendum.
Previous first round winners include Mr Clarke in 1997, Michael Portillo in 2001 and David Davis in 2005, none of whom went on to actually become Tory leader.
He said he had been “overwhelmed” by colleagues’ support.
She is regarded as the strongest leader, the most unifying candidate, the candidate with the best economic understanding and the best media performer by Conservative party members.
The bookmakers also predict that it will be May who is most likely to be the next British prime minister.
Asked if he thought the contest could be settled before 9 September, he said: “That will be for others to decide”. “As long as Gove doesn’t come in the final two I don’t mind what happens”, Rifkind said to Clarke.
Michael Gove was instrumental in seeing Johnson not take the top job.
“And for that reason I’ve chose to give my support to Theresa May – I intend to work closely with her, to campaign for her, and I’m sure she will be a very fine prime minister of this country”.
Former London mayor Boris Johnson has thrown his weight behind Andrea Leadsom in the Conservative Party leadership contest.
Mr Johnson subsequently announced he would be ruling himself out of the race.
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U.K. Home Secretary Theresa May campaigned for the U.K.to remain in the European Union alongside her soon to be ex-boss, Prime Minister David Cameron.
