The move follows Adidas' acquisition of the rival Reebok company last year.
Carat's two teams, the Adidas one based in London and Reebok's in Boston in the US, will be co-ordinated by Nigel Morris, the chief executive of Aegis' digital unit, Isobar. Morris has worked on Adidas for more than six years.
The move follows Carat's triumph over OMD in a final pitch after Starcom and MediaCom had earlier withdrawn.
In the UK, the decision means the £6 million Adidas and £4 million Reebok accounts move from PHD and OMD respectively into Carat.
Before the pitch, Carat handled Adidas globally, with the exception of the UK. OMD handled Reebok across Europe.
Uli Becker, Adidas' head of global brand marketing, said Carat's "strong strategic vision and commitment to our business" made it the winner.