Strong favourite Theresa May topped Tuesday's first poll of Conservative lawmakers in the race to succeed Prime Minister David Cameron.

Home Secretary May won 165 votes in the poll of 329 lawmakers, followed by Vote Leave campaigner Andrea Leadsom with 66 votes.

The election of a new leader was prompted by Cameron's resignation following his failure to persuade a majority of voters in a June 23 referendum to back his call for Britain to remain in the European Union.

Pro-Brexit candidate Liam Fox, a former defence secretary, was eliminated from the leadership contest under party rules after he gained the lowest number of votes, 16.

The remaining four candidates qualify for a second ballot by lawmakers on Thursday.

May received another boost late Tuesday when pro-Remain Welfare Secretary Stephen Crabb, who 34 got votes on Tuesday, tweeted that he was dropping out to support May.

"Following tonight's vote I will not be putting my name forward for the next round of #toryleadership - and will be backing @TheresaMay2016," Crabb said.

In a further, indirect boost for May, two veteran Conservative politicians appeared to back her in unguarded remarks captured by an open microphone.

"Theresa is a bloody difficult woman, but you and I worked with Margaret Thatcher," Ken Clarke, a former chancellor and home secretary, laughed as he spoke to another senior Conservative off-air in a Sky News television studio.

"I get on all right with her [May] ... and she is good," Clarke, 76, told former defence minister Malcolm Rifkind while still wearing a microphone following an interview.

Rifkind said he was also "impressed" with May, who is the clear favourite to become Britain's next leader.

Clarke claimed that if pro-Brexit Justice Secretary Michael Gove - one of the other three remaining candidates - became prime minister "we'd go to war with at least three countries at once."

But he said Gove, who got 48 votes, "did us all a favour" by forcing his former Vote Leave co-leader, Boris Johnson, to quit the race.

"The idea of Boris as prime minister is ridiculous," Clarke said in his remarks, which were also captured by a Sky News camera.

He said he believes that neither Johnson nor Leadsom, who is now backed by Johnson to lead the party, "actually are in favour of leaving the European Union."

May supported Cameron's Remain campaign but kept a low profile in the final weeks before the June 23 referendum. She has vowed to provide "stability and certainty" to Britain and to unify her divided party.

Three of the other four candidates backed Brexit, and some pro-Brexit politicians inside and outside the party have said the next leader must be someone with a strong commitment to taking Britain out of the EU.

Gove made a surprise last-minute decision to stand after saying he had lost faith in Johnson's leadership ability.

Johnson then dropped out of the race, later giving his support to Leadsom, who had appeared alongside him in the biggest television debate.

After Thursday's vote, one more round of voting by lawmakers will follow to produce a shortlist of two candidates for a vote by the party's 150,000 members.

May is expected to be in the final vote against one of her two remaining pro-Brexit rivals, Leadsom or Gove.

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