The 56-year-old, who became a stockbroker following his retirement in 1991, will take up the role with the ECB on February 1.
"I am honoured to have been offered this position and the trust which the ECB has placed in me to lead our national teams," Downton said in an official statement.
"Since retiring from the game I have taken a close interest in both county and international cricket and I am excited by the challenge of building on the success which England teams have enjoyed at all levels in recent years."
Morris, 49, announced in August that he would be stepping down in order to join Glamorgan as director of cricket.
The former batsman joined the ECB in 1997.