Morgan's appointment was confirmed at a press conference at The Oval following a two-day meeting of the ICC's executive board in London.
The former chairman of Glamorgan, the one Welsh side in England's 18 first-class counties, will replace Ray Mali who has been acting president since fellow South African Percy Sonn died last month.
Morgan will take over from Mali in May next year.
Initially, the ICC board had been split between Morgan and India's Sharad Pawar over who should succeed Sonn.
But, in what appeared to be a compromise decision, Morgan will only be in the post for two years with Pawar due to replace him in 2010 before he too serves a two-year term ending in 2012.
The ICC announced that from 2008, all presidential terms would be limited to two years with no option for a third year in office.
Morgan is now set to relinquish his role within the ECB and Pawar is also due to step down from his position as president of the Board of Control for Cricket in India within the next 12 months.
"I am delighted and greatly honoured to take on the role of ICC president-elect and, from June 2008, the ICC Presidency," said Morgan in an ICC statement.
"My immediate priority over the next few weeks is to complete unfinished work with the England and Wales Cricket Board (ECB) and also to discuss with the ECB an orderly and timely process for the election of my successor as chairman.
"Thereafter my priority will be to support current president Ray Mali for the remainder of his term in office.
"I am hugely grateful to Ray because it was one year ago that, as president of Cricket South Africa, he made the first approach to me to become a candidate for the role of ICC president."
Mali said: "We now have certainty in terms of who will head up our game for the next five years."
Pawar, who thanked ICC members for their "confidence" in him, added: "I am particularly happy with the approach taken by the governance review committee in looking to resolve the issue of the succession and I believe David Morgan and myself have readily accepted those proposals in a sporting manner."