The royal was speaking as she visited a school in Hayle, Cornwall with United States First Lady Jill Biden at the G7 summit.
Harry and Meghan announced the birth of their the girl, who was born on June 4, last Sunday.
Kate told reporters: “I wish her all the best, I can’t wait to meet her because we haven’t met her yet, so I hope it will be soon. “
The Duchess and Ms Biden were visiting an academy for children ages four to 11 to highlight the issue of early childhood development in the early years.
This is the first time that the two women meet and during the visit they organized a roundtable on preschool education.
They also visited a classroom, meeting some of the youngest children at Connor Downs Academy.
They also brought carrots which were fed by the children to the school pet rabbits in the park.
Ms Biden was asked to meet with the Royal Family on Thursday – along with the Bidens who were due to have tea with the Queen at Windsor Castle on Sunday.
She replied, “Well, we hope to focus, I think, on education.
“Joe and I are both looking forward to meeting the Queen. It is an exciting part of the visit for us. We’ve been looking forward to this for weeks and now it’s finally here. It’s a great start. “
Ms Biden and Boris Johnson’s wife Carrie Johnson enjoyed the view over Carbis Bay on Thursday before politicians hold their first summit meeting.
The Duke and Duchess of Cambridge are starting to make their mark as members of the Royal Family with a number of high profile events at the summit.
William will later join his father in hosting a reception for executives and CEOs of some of the world’s largest companies to discuss how the private sector can work with governments to tackle the climate emergency.
The couple will then join the Queen, Prince of Wales and Duchess of Cornwall at the G7 leaders’ reception this evening.