Jeppesen is step down in August to return to the US. It is not known whether he has a job to go to.
Burns moves from his current role as director of response based marketing.
He joined Barclays last year from US bank Washington Mutual, where he was senior vice president, direct marketing. He also previously worked at Procter & Gamble as a brand manager.
Jeppesen has been in his current role since 2007 after taking over from Jim Hytner. He was previously head of direct response at the bank.
He was involved in the retail bank’s latest ‘one small step’ campaign that aims to reposition Barclays as a bank that can help customers stay in control of their personal finances
Burns will head up all UK retail banking marketing activity including advertising and in branch merchandising across all retail bank products and distribution channels.
He will report to global retail and commercial banking chief marketing officer Libby Chambers.