The appointment ends a 13-year relationship with BMP, which, over the years, has created famous advertising for the brand including the "Latham" campaign starring Rowan Atkinson.
The move completes the consolidation of all Barclays' above-the-line creative advertising business into BBH.
Both agencies were called in to give presentations at the end of last week. Barclays has not briefed BBH to produce a campaign.
Barclays said the move was part of the bank's ongoing strategy to create a unified voice for the group.
"We are keen to build on the brand's strength through a fresh approach, helping us to develop new links with existing and potential customers. It is for this reason that BBH is going to take the account forward," Sara Deeks, Barclays' brand strategy director, said.
"BBH has demonstrated a great understanding of our needs in its other work across the group's portfolio, including Barclays main 'fluent in finance' brand campaign, work on The Woolwich and for Barclays Private Clients."
"BMP has done an excellent job for Barclaycard over the past few years," Deeks added. " It has helped establish Barclaycard as the leading credit card brand in the UK."
BMP won the account from Collett Dickenson Pearce in 1990. The agency dropped Alan Whicker, replacing him with Atkinson's MI6 agent. Later, Angus Deayton became the face of Barclaycard.
Earlier this year, Barclaycard moved its media account out of OMD UK and into Starcom Motive. BMP's fellow Omnicom agency WWAV Rapp Collins, which handles Barclaycard's below-the-line account, is unaffected.