The Bank won the business, for which there was no incumbent, after a competitive pitch run directly by the brand.
The £2.5 million campaign is due to launch later this month and will introduce a new range of chilled desserts by Bonne Maman, which is owned by the Andros Group.
MediaCom handles Bonne Maman's media planning and buying and was unaffected by the review.
Tristan Connell, the managing director at The Bank, said: "Bonne Maman is a beloved brand with a clear ambition to strengthen its position in the UK and reach a larger audience.
"We are looking forward to inspiring UK consumers with our new campaign."