He will take up his new post in October, replacing Derek Purnell, who stepped down last year to become an arts consultant.
Barron is the third top official to leave the beleaguered Scottish Opera in the last year. Chairman Duncan McGhie stepped down last year, and music director Richard Armstrong announced in December that he would leave in July.
Barron's tenure was marred by Scottish Opera's financial troubles, which culminated in the decision last year to lay off 88 staff and chorus members and shut down for a year starting this spring.
According to press reports, Barron's departure was expected, both because of the Scottish Opera's struggles and because the opera and ballet companies recently decided to split their boards, which had been merged in 1999.
Before coming to Scottish Ballet and Opera, Barron was artistic director and chief executive of the Brighton Festival and Dome. Previously, he worked at the Edinburgh Festival and the Buxton Festival, among other arts organizations.