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Andrea Leadsom, one of the Conservative candidates running to be the next prime minister, said she switched her stance on the EU in response to a change in the facts.
The MP - who previously supported the UK remaining in the bloc - said she reconsidered after David Cameron failed to secure major changes in his efforts to renegotiate the relationship.
Stephen Crabb has called the Brexit vote to leave the EU a "new chance", but warned a "big national push" would be needed.
In an interview with The Times, the Remain campaigner who has thrown his hat in the ring for the Tory leadership, said the "out" vote would be implemented, but it was important to first establish the "overriding principles" that would guide negotiations.
He said "it is essential" the party showed "real substantial change on immigration" as part of that process.
As the Tory leadership contest hots up, Theresa May appears to be in the lead.
ITV News understands the home secretary can so far count on the backing of at least 82 Conservative MPs, putting her quite a way ahead of her rivals.
ITV News Deputy Political Editor Chris Ship reports.
Latest ITV News reports
Michael Gove has become the first of the Conservative leadership candidates to publish his tax returns during the campaign.
Theresa May has secured the backing of Philip Hammond in her bid to succeed David Cameron ahead of the first round of voting on Tuesday.