Their itinerary for the visit includes engagements in Fiji and Tonga, countries which have been classified as having a moderate risk of Zika virus transmission.
As the Duke and Duchess of Sussex made their way past hundreds of fans, a cheer could be heard when Prince Harry spotted Mrs Dunne.
Meghan and Harry are now in Australia on an official 16-day tour and will be travelling to Fiji, Tonga and New Zealand.
Meghan Markle is pregnant, Kensington Palace announced on Monday, and will give birth next year. With Meghan's darker features mixed with Harry's lighter ones, Joe predicts that a baby girl would have dark hair and brown eyes, while a boy would have lighter brown hair and blue eyes.
Meghan, with her American roots and successful acting career, has been seen as a modernizing influence on the sometimes stodgy royal family, and she is credited by many for bringing happiness to Harry, who has long struggled to cope with the early death of his mother, Princess Diana.More news: Malicious Message Bug Is Crashing PlayStation 4, How To Protect Your Console
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Prince Harry - who has previously met koalas - was a tad more irreverent during the meeting.
Two children, selected by the zoo through its Youth Wild Squad, approached the royals to give them a bouquet of flowers.
The couple posed for a photograph on the croquet lawn with the Governor General and his wife with the magnificent backdrop of Sydney Opera House behind them.
Meghan chose the fitted shift dress as she stepped out in public for the first time since the news of her pregnancy was revealed.
Governor-General Sir Peter Cosgrove, the queen's representative in Australia, presented the couple with their first baby gifts, a stuffed kangaroo and a tiny pair of fleece-lined Ugg boots.
She captivated Britain. There was an ease to her confident smile, and contagious excitement about moving across the Atlantic and joining the royal family.