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The musician, who rose to fame as a member of the boyband One Direction, was scheduled to play one show at Auckland's Spark Arena on November 23.
Styles posted on his Twitter account overnight: "Everyone's health and safety remains our top priority, which is why I unfortunately have to postpone all 2020 shows in South America, Mexico, Australia and New Zealand until further notice.
While he was not announcing new dates just yet, Styles added that he was "closely monitoring" when it would be safe to tour again.
"I really hope to play the shows as planned for 2021 but will continue to monitor the situation over the coming weeks and months. I can't wait to see you all on the road as soon as it's safe to do so."
Live Nation added their own statement on social media, saying the dates would be rescheduled soon.
"The rescheduled dates will be announced in due course. Purchased tickets for the original dates will be valid for the rescheduled dates. Ticketholders are encouraged to hang onto their tickets until dates have been confirmed. For more info please contact your point of purchase."
Styles last played a show in New Zealand in 2018. The tour was in support of his latest album Fine Line which features hits Lights Up, Adore You and Watermelon Sugar.
He became the first UK male artist to debut at #1 in the US with his first two albums. It topped the Billboard album charts when it was released and broke records for the biggest sales week for a UK male artist since Nielsen Music began electronically tracking sales data in 1991.