The utility group - owner of Anglian Water - said it was selling the Morrison construction services and project investments businesses to Galliford Try for 42m. The deal will also include two early-stage PFI (public finance initiative) projects.
Morrison Construction Services employs around 1500 staff at eight main offices throughout the UK.
Based in Edinburgh, the project investments arm has 15 employees and focuses on bidding for selective PFI projects.
Commenting on today's deal, which still requires shareholders' consent, AWG chief executive Jonson Cox said: "Two years ago, faced with an unprofitable construction business, we set out to re-build the performance of Morrison Construction Services.
"The sale is a very good result for all parties. For shareholders, it delivers good value and focuses AWG's business activities on water and support services. For employees, it secures a strong future as part of a group with which there is a good cultural and business fit."
Last month, AWG and Sir Fraser Morrison settled their long-running legal battle, a week before the dispute was due to reach London's High Court. The proceedings centred on claims that Morrison, which was founded by Sir Fraser, misrepresented its true financial position when AWG bought it in 2000.
AWG was seeking 130m from Sir Fraser and Stephen McBrierty, Morrison's ex-head of construction - the difference between the price it paid and what it claimed was the company's true value.
Today's deal does not include Morrison Utility Services and Morrison Facilities Services.