RMJM chief executive Brian Stewart, who led the management team, said Morrison's stake had not been finalised, but that it would be a substantial minority holding. RMJM's banker, the Bank of Scotland, has maintained a 6% stake with the management team holding the rest.
Morrison has kept a low profile since he and his brother Gordon sold their family contracting firm to utilities company AWG for £262.5m in 2000.
AWG has not ruled out taking legal action against the Morrisons after it discovered the business was not worth as much as it had thought.
Last week, AWG said the Scottish-based Morrison business would cost it £15m this year in addition to the £57m it had already lost on the firm.
Morrison replaces Mark Way as chairman of RMJM. Way has become head of the firm's research and development department. Stewart said RMJM was pleased to have Morrison on the board because he had a reputation as a leader of change.
Stewart said the management takeover was essential for the firm's growth.
He said: "Our growth was impeded because more than 60% of the shareholding was outside the company and this was increasing as more people retired. The deal gives us more scope to do the things we want to do and the cash that has come in will also be used to grow."
RMJM has four offices in England and Scotland as well as offices in Dubai, Singapore, Manila and Bangkok.