The announcement comes as Harry and Meghan landed in Sydney to begin their 16-day tour of Australia, New Zealand and the Pacific.
The palace said the couple had "appreciated all of the support they have received from people around the world since their wedding in May and are delighted to be able to share this happy news with the public".
The Duke and Duchess of Sussex arrived in a wet and wintery Australia after a weekend spent celebrating the marriage of their relative Princess Eugenie to a wine merchant.
The couple, who married in May, were beaming with happiness as they kicked off their first global tour as a married couple in Sydney, Australia on Tuesday.
Living close by together at Kensington Palace, Prince George, Princess Charlotte and Prince Louis are likely to spend a great deal of time with their new baby cousin.
The tour coincides with the Invictus Games in Sydney which start on Saturday.
The Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall and Duke and Duchess of Cambridge are said to be "delighted" for the couple.
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The couple were given the first toys for the new arrival, with the parents-to-be handed a soft kangaroo and joey by his Excellency General the Honourable Sir Peter Cosgrove and his wife, Lady Cosgrove, at a welcoming ceremony at their Admiralty House residence in Sydney, overlooking the harbour.
Harry and Meghan's child will not, however, be in immediate line to the throne.
After the speech, Meghan spoke to guests including Australian TV personality Eddie Woo, who said his own children - aged five, seven and ten - require "a lot of energy", to which she replied: "We're ready! I wish them both good luck".
The Duke and Duchess kicked off their Royal tour of Australia just hours after Kensington Palace confirmed the Duchess of Sussex was expecting her first baby.
A doting grandfather, the prince will want to pass on his love of the outdoors and the natural world to his new grandchild, and read him or her the stories he enjoyed as a child.
Then, as if out of nowhere, a cheeky-looking Prince Harry storms towards the person he assumes gave his wife the bunch of flowers and retorts: "You can't give flowers that big to my wife, what is that all about!"
President Donald Trump's representative in Britain, U.S. Ambassador Woody Johnson, tweeted: "Happy news to wake up to on a Monday morning - congratulations to the Duke and Duchess of Sussex!".