Snowball, the agency's existing chief executive, will become the chairman, while Peter Souter remains as the executive creative director.
However, Baulk will remain as the group chairman and chief executive, also overseeing Proximity and Redwood. The ad agency accounts for 30% of the UK group's revenue.
Baulk said: "It's a combination of me being ready and having a group of people to take the agency forward. I have no doubt it's the right thing to do and I have every trust in Cilla as my successor."
He added: "Farah's intelligence, energy and passion for success will bring a formidable drive to the agency's agenda."
However, a run of difficult years has seen the agency struggling to hold on to its position as the market leader. Snowball said: "We know we have to implement change and continue to grow."
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