As a left-handed opener, Morris twice captained the Welsh county side and helped them win the 1997 County Championship title.
He retired that year to join the ECB, where he has also held the positions of technical coaching director, acting chief executive and deputy chief executive.
He was appointed as managing director in 2007 and has helped England win the last three Ashes series.
"This has been a remarkable period in the history of cricket in England and Wales and I have been extremely proud to having been able to play a part in it,” Morris said.
"In my time at the ECB I have been fortunate to work with some extremely dedicated and talented cricketing people and I thank them and the Board all for the support they have given me and their contribution to the success of the England teams.
"It has been my privilege to work with Andy Flower, the England team director, during the last four years and congratulate the players on their three Ashes series successes, being the number one ranked team in the world in all three formats of the game, and also winning one ICC Global event and being runners-up in another.
"I am now looking forward to a new and exciting challenge with Glamorgan which I will tackle with energy, enthusiasm and relish. I wish the England teams at all levels and the ECB the very best of luck in the coming years and I am sure they will enjoy further success."
David Collier, the ECB chief exec, added: "Hugh has been an influential figure within the ECB since its inception in 1997 and his vision has helped deliver this outstandingly successful period in our cricketing history.
"He will be sadly missed at Lord’s by all the ECB staff and at Loughborough where he provided the blueprint for the widely admired National Cricket Performance Centre (which launched in 2003). We all wish him well on this latest phase of his career with Glamorgan County Cricket Club.
"We will now begin a recruitment process for a replacement and I am confident that we will attract the calibre of person who can continue and build on England’s successes at all levels."
Alan Hamer will resign his role of chief executive at Glamorgan at the end of the season, with Morris, 49, then set to take over.
Glamorgan cricket chairman Barry O'Brien said: "Alan has indicated to me for a while that he wishes to pursue a number of other opportunities.
"As our previous finance and commercial director and chief executive since 2009, Alan has made a major contribution to the club, including managing the successful [hosting of] 2009 Ashes and this year's ICC Champions Trophy.
"I am very grateful to Alan and respect his decision to seek a new challenge."
Hamer said: "This has been a difficult decision following seven happy years at the club.
"I am, however, convinced that this is the right time for me to build on my successes at the club and to take on a new challenge."