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A YouGov 'on-the-day poll' suggests 52% for Remain and 48% for Leave.
Chancellor George Osborne has said the vote to leave the EU was "not the outcome I wanted" but that he will "do all I can to make it work".
Mr Osborne, who campaigned for remain, described it as a "hard fought campaign".
He said he had discussed the decision with G7 finance ministers and bank governors.
Ex-Welsh Secretary David Jones MP returns to government as Theresa May's cabinet reshuffle continues.
The Clwyd West MP has been appointed as minister of state in the Department for Exiting the European Union.
David Jones led the Vote Leave campaign in Wales and will now assist David Davis in negotiation to leave the EU.
Latest ITV News reports
David Cameron has announced he is to resign as prime minister by October. We look at who could replace him.
We look at the implications of the EU referendum result on various aspects of life in Britain.