The struggling mutual will use the proceeds to pay down debt and invest in its remaining operations, which include food shops, funeralcare and insurance.
The deal involves 15 farms, including three packing sites, and almost 40,000 acres of land. The Co-op said it was pleased to have found a buyer that will continue to invest and develop the farms business, which dates back to 1896.
The sites grow a variety of cereals, including wheat for bread flour, and a number of fruit and vegetables, such as potatoes, cider apples and peas.
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However, the enterprise was not considered a core business to the Co-op as it supplies only a small proportion of the food sold in its stores. The move follows last month's sale of its pharmacy stores to Bestway for £620 million.
The Wellcome Trust, which is the world's second highest-spending charitable foundation, has a property estate valued at around £1.8 billion.
He added: "The Trust's philosophy is to provide long-term investment for the businesses and property we hold in good times and in bad.
"We will take this approach to running Farmcare Trading as a going concern, giving a business that is already strong and successful the support and resources it needs to grow, to the benefit of employees, tenants, partners and local communities."
The Trust has pledged to maintain existing services for local communities, such as the educational programme for children - "Farm to Fork".
The sale includes more than 100 residential properties and 27 commercial buildings. The farms are at Aberdeenshire, Ashby St Ledgers, Blairgowrie, Borders, Carnoustie, Coldham, Down Ampney, Goole, Highland Court, Langley Brook, Longforgan, Louth, Rockingham, Stoughton, Tillington and Wykeham.
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