Dacre will be stepping down from the post after 28 years in September.
Greig (pictured) has been editor of The Mail on Sunday for six years. He was previously editor at the London Evening Standard in 2009, and has also worked for The Sunday Times.
Ted Verity, deputy editor at the Daily Mail, will take over at The Mail on Sunday.
Lord Rothermere, chairman of the paper’s publisher DMGT, said: "Geordie has been an outstanding editor of The Mail on Sunday, and I am delighted that he will continue the high-quality journalism that Paul has made a hallmark of the Daily Mail for more than 25 years."
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