BBH saw off competition from Britvic's roster agency Clemmow Hornby Inge as well as Delaney Lund Knox Warren & Partners and BMP DDB to win the business, following a pitch overseen by Agency Insight.
BBH will begin working on the account from 1 August, with a brief to create a new campaign for the brand to air in 2004.
The account was previously held by HHCL/Red Cell. The incumbent resigned the business after eight years in April, a decision motivated by the Red Cell network's links to Coca-Cola in the US.
BBH's joint chief chairman Nigel Bogle said: "We are delighted to be working on a brand that combines such a strong heritage with an innovative and forward-thinking approach to marketing."
Britvic's category director, Andrew Marsden, said: "All the pitch agencies responded well to the brief, but BBH demonstrated a highly creative approach that captured the essence of Robinsons. It also has a strong track record of working with household brands and I am looking forward to a successful working partnership."
The win caps a good few months for BBH in terms of new business. Over the past month, the agency has pulled in both the Bisto account and an anti-smoking brief from Cancer Research. This is in addition to the Woolworths business it won from Bates earlier this year.
The appointment does not affect Britvic's relationship with its direct marketing agency, 141, or its media agency, MindShare.
Last month, Britvic launched a £3 million through-the-line campaign to drive sales of the Robinsons High Juice drink throughout the summer.
A 141-created below-the-line initiative consisting of door drops and sampling was accompanied by HHCL print work. The campaign tied in with the re-release of the Robinsons' summer flavour Tropical Fruits.