Martin was previously deputy managing director of the agency, having been promoted from managing partner in March 2017. She will continue to oversee the Tesco business alongside her management responsibilities.
BBH says Arnold will work alongside Neil Munn, the global chief executive of BBH and worldwide chief creative officer Pelle Sjoenell on building its global brand. He will continue to run the UK agency’s clients Absolut, Audi and Google.
Munn said: "We decided in 2017 that we needed a global chief marketing officer to really own and drive the business.
"We have been looking for over six months for a leader with both the skill set and a robust understanding of the BBH brand. Adam Arnold is that person and his energy, intellect and proactivity will be key in this new role".
Arnold was made UK MD in May 2016, replacing Mel Exon who joined The Sunshine Company as chief executive.
Ben Fennell, the chief executive of BBH London, added: "Once again BBH has promoted from within. I am delighted that Karen will be taking BBH forward. Karen’s infectious energy and personality, coupled with her commercial and creative ambition, will continue to see BBH London grow from strength to strength."