Greig's appointment follows the announcement on Wednesday that Paul Dacre was to step down from his role as editor in November to become the chairman and editor-in-chief of the title's publisher, Associated Newspapers.
Ted Verity, currently deputy editor at the Daily Mail, will be the new editor of The Mail on Sunday, DMGT said in a statement.
(Reporting by Shalini Nagarajan in Bengaluru; Editing by Alexandra Hudson)