Burton will join Drury Porter Novelli, which is fully owned by Omnicom, in the New Year as head of corporate.
Before last year's EU referendum, it was argued that Britain might lose out to Ireland if it left the EU, because multinational companies might become more likely to put regional headquarters or hubs in Dublin rather than London. This narrative has continued since the vote, with several companies reported to have made decisions along these lines.
It is understood that Weber will now split the role and recruit for on an MD of its corporate practice and another for its public affairs division, with the latter including recently hired former Downing Street staffers Joey Jones and Kate Shouesmith.
Both sectors are overseen by chair Jon McLeod, who is also in charge of Weber's Manchester business.
London MD Rachel Friend said: "Ben has always been a highly collaborative leader across our London office, a great client counsellor, and a good friend to many team members past and present.
"In the short term the existing corporate, financial and public affairs senior team will take on combined leadership and client responsibilities."
Weber Shandwick will say goodbye to its EMEA CEO Colin Byrne in 2018, with current EMEA and APAC chair Tim Sutton moving into his role.