Once released from RSA, Adrian will become executive chairman, underwriting and distribution at expanding broker Arthur J Gallagher.
"We have an ambitious business plan which now needs to be driven over a time horizon that Adrian could not reasonably commit to," said Stephen Hester, RSA chief executive.
"Adrian has served RSA with distinction for 25 years, including since 2008 as CEO of our UK business. He has many friends here and in the industry and we are grateful for all he has done for RSA.
"We will miss him when he leaves but look forward to a strong relationship continuing at his new employer," said Hester.
Brown said: "RSA has been my home for 25 years. I have enormous affection for the group and confidence in its future.
"I will remain fully engaged in the coming weeks with business as usual and in due course look forward to continuing a strong relationship with RSA in my new role."
At Arthur J Gallagher Brown will report directly to David Ross, the AJG international CEO.
David Ross said: “Recruiting Adrian is a huge hire for us.
“At a time of our own transformational change — most notably in our UK commercial broking division, following the recent acquisitions of Giles and Oval — the opportunities for expansion and growth within personal lines broking and underwriting remain substantial.”
Adrian will take over the leadership of underwriting and distribution from John Hastings-Bass, who will be Asia-Pacific chairman, focusing on developing international business opportunities outside the UK. Hastings-Bass will be working with chairman Tom Gallagher, David Ross and International Commercial Director Vyvienne Wade.