Kensington Palace said the prince added: "I can't wait to meet her".
Harry (30) is currently on military duty in Australia, his last assignment before he resigns from the military service in the UK.
It's understood he will remain in Australia for another three weeks.
Read more here: Royal Baby: Pippa and Carole Middleton arrive to visit the princess
Harry was one of the first members of the royal family to be informed of his niece's birth.
The Princess of Cambridge - as she has been dubbed until her parents make her name public - made her arrival just after 8.30am yesterday morning.
She was born to mum Kate Middleton at the Lindo Wing of St Mary's Maternity Hospital in London. Her father, Prince William, was by her mother's side.
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Her big brother - Prince George (22 months) - met his little sister for the first time in the hospital yesterday afternoon.
Kate was admitted to the hospital at 6am yesterday and gave birth just two and a half hours later.
At 6.10pm yesterday evening - just over 12 hours since she was admitted - she left the hospital and showed the beautiful baby princess to the world.
Kate, William, George and the baby girl spent their first night together as a family last night in Kensington Palace.
Today, Kate's parents, Carole and Michael, her sister Pippa, and William's father Prince Charles and his wife Camilla visited the palace and met the baby girl for the first time.
Earlier this afternoon, Kensington Palace tweeted on their official account that Kate and William are "hugely grateful for the messages of congratulations from around the world.
"It means a great deal to them that so many people have celebrated the arrival of their new daughter."
Read more here: #WelcomeToTheFamily - Royal household officially welcomes newborn princess on Twitter