Barrett is currently editor of Haymarket title Young People Now, which is aimed at youth workers, and has also worked on digital media business title Revolution as associate editor.
Before becoming a journalist he was sales and marketing director of Leeds-based digital media agency Brand New Media, whose clients included Heinz, Barr Soft Drinks and KP Foods.
Barrett will take up his new position in November, and replaces Philip Smith, who recently moved to Brand Republic, Haymarket's marketing and media news website, in the newly-created role of head of content.
"I am looking forward to the challenge of making Media Week an even stronger brand than it already is," said Barrett. "The media world is undergoing tremendous changes and I am inheriting a strong team and a well-respected title that can help its readers navigate through this new landscape."
Colin Grimshaw, Media Week's deputy editor, will remain as acting editor until Barrett's arrival.