He is described as "The Don" of advertising and marketing law by the 2005 Chambers Directory, which ranks Britain's specialist legal experts.
"The firm's technical proficiency and commercial acumen is second to none," according to clients polled by Chambers.
Alexander, 62, has been a senior partner at the 43-partner firm since 1999 and has been associated with some of the biggest agency deals in recent decades. He advised Abbott Mead Vickers from its inception to its eventual sale in 1999 to Omnicom Group for £360 million.
He also defended Christine Walker against her old company, Zenith, in her successful battle to set up Walker Media.
Ian Jeffrey, 37, an intellectual property lawyer in the firm's new Media, Brands and Technology group, is taking over the role of managing partner.
Trevor Watkins vacates the position and will continue to practise in the firm's corporate department.
Alexander said the move is intended to build the quality of the Lewis Silkin team and allow a new generation of young managers to shine. "The new role will allow me to play an even greater role in relation to our major clients," he said.
Brinsley Dresden is also promoted to run the Media, Brands and Technology department and Jo Evans takes over as the head of the firm's corporate department.