Its name originated from the fact that it served as the residence for the Commander-in-Chief of the Royal Navy's Australia Squadron from 1885 to 1913.
The couple who married in late May were seen holding hands as they stepped off the plane, with Megan pictured beaming. The duchess, who hasn't been to Australia before, marvelled at the view.
They will then travel by boat across to the Opera House where they will watch a rehearsal by the Bangarra Dance Theatre before fronting the crowds.
On a trip to Belfast less than two months before she married Harry in May, the former actress joked when she was shown an innovative range for newborns: "I'm sure at some point we'll need the whole (lot)". She is due in the spring of 2019, less than one year before her and Harry's one-year wedding anniversary.
The couple were given the toy and a pair of tiny Ugg boots as souvenirs from Australia.
"It's somewhere he feels very relaxed", Jackson said.
"These are awesome", said Harry as he posed for the cameras holding them.
"When you go to Dubbo you'll see some of these", said Sir Peter. Meghan replied: "Thank you so much".
An official outside the Opera House told 9Honey the 37-year-old expectant mum had to cut the meet-and-greet short because she was feeling unwell.
"I think it could possibly be a girl".
The games were founded by Prince Harry are an worldwide paralympic-style event for military personnel wounded in action.
While at the zoo, the couple ticked off a very important Aussie bucket list item: meeting a koala and her newborn joeys.
But the youngster will still be behind his cousins Prince George and Princess Charlotte, by virtue of their father - Prince William - being older than Harry and second in line to the throne.
The visit to Australia is their first overseas tour since Harry, 34, married the USA television star Meghan Markle in May.
The visit coincides with the Invictus Games in Sydney, which kick off on Saturday and run for a week. The games founded by Harry are an global paralympic-style event for military personnel wounded in action.
Harry and Meghan said during a television interview in November that they wanted to promote humanitarian causes close to their hearts across Commonwealth-member countries, including Australia.
A spokesperson also revealed that the Duke and Duchess of Cambridge "are delighted for the couple and were able to congratulate them on Friday at the wedding in person".
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