The business was estimated to be worth £20m when BBH won the account amid declining sales for the Smirnoff Ice brand.
The switch consolidates the entire Smirnoff business within JWT.
The change is understood to have been switched after creative presentations to Diageo marketing chiefs from both agencies.
JWT had previously worked on Smirnoff Ice for four years and created the "As clear as your conscience" campaign, which was criticised for its sexual undertones.
BBH's first work for Smirnoff Ice featured an eccentric Russian character called Uri and was targeted at 18- to 24-year-old men.
For the full story, read tomorrow's edition of Campaign.
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