The position has been vacant since 2011, when Paul Lawson was promoted to chief executive.
Lee has been at the company for seven years, most recently leading Leo Burnett Group’s business development and communications.
She will retain these responsibilities in her new role.
Lawson said: "Heading up our group marketing, Katie has brought a focus, drive and discipline to our new-business and reputational success. I know that her ‘can-do’ mentality and belief in the Leo Burnett brand is absolutely right for her role as managing director. For me, there really is no-one better for the job."
Lee has more than 15 years’ experience in advertising. She joined Leo Burnett in 2007 as a client service director to work on McDonald’s and has also led government work and the Homebase account.
She was promoted to deputy head of account management in 2010, which included responsibility for all agency training and graduate recruitment, and was appointed to the group marketing director role in 2011.
Lee said: "We’ve spent the last few years looking at the Leo Burnett Group from an integrated perspective, both in our external offering and our internal culture.
"Having a managing director for Leo Burnett, the ad agency, allows us to bring the focus back to a brand with many deeply talented people behind it."
Lee is the joint chair of the IPA’s New Business and Marketing Group. She began her career at CDP and has also worked at Saatchi & Saatchi and Bartle Bogle Hegarty.