He succeeds Michael Cudlipp, who died in October last year aged 70, having built HAT into the largest archive of its kind in the world.
Cox's first major task will be to take over from Cudlipp in overseeing the construction of a £100,000 extension to HAT's headquarters in Raveningham, near Norwich, which is due for completion in the autumn.
A one-time JWT trainee, Cox worked for Collett Dickenson Pearce before joining Publicis in the early 90s. He became the head of business development in 1992 before taking over as the managing partner. He was also closely involved in Guinness' launch of the Enigma lager brand in the UK in 1995.
He joined Grey in 1997, taking responsibility for clients including GlaxoSmithKline, Boots, United Biscuits and COI Communications.
Cox said: "What the team here has achieved is quite remarkable. The next stage of the development offers us all the opportunity to put HAT on the road to becoming a centre of study of national importance for all those studying advertising, marketing and media subjects."