The 59-year-old served as CEO of De Beers from 2013 until early 2023 and was co-chair for a further year after that.
Cleaver replaces Martin Tolcher, who was chair of Gemfields for five years and has been on its board for the past 15 years.
In a statement, Cleaver said that there were significant similarities between De Beers and Gemfields.
“Gemfields is a unique and extraordinary business, one that has taken a fragmented and fairly informal industry to considerable sophistication as the first consistent supplier of significant quantities of coloured gemstones in a responsible and transparent manner,” Cleaver said.
“The parallels with De Beers’ origins and how consistent and reliable supply can deliver remarkable industry growth and positive contributions to communities, are clear to all. The coloured gemstone market has long-transcended the arrival of their lab-grown counterparts, with lab-grown rubies having been around for more than 120 years.”
Specialising in the mining of colour gemstones, the company accounts for around 25 per cent of the emerald supply, and a further 50 per cent of the ruby supply worldwide.
Gemfields also named two new board directors: Simon Scott, who previously served as CFO of Lonmin, a platinum miner; and Kieran Daly, managing director of Assore International Holdings.
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