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Barclay, a commercial lawyer and director of New Zealand Cricket since 2012, replaces interim chairman Imran Khwaja who himself had stepped in to temporarily succeed India's Shashank Manohar after his second two-year term ended in July.
"It is an honour to be elected as the Chair of the International Cricket Council and I would like to thank my fellow ICC Directors for their support," Barclay said in a statement.
"I hope we can come together to lead the sport and emerge from the global pandemic in a strong position and poised for growth."
Barclay was NZC's representative on the ICC board, but will step down from his role with NZC to lead the global governing body.
He was also a director of the men's World Cup that was jointly held by Australia and New Zealand in 2015.