Beatrice had been due to give birth in early Spring, but in December had received medical advice not to travel long distances.
The photo issued of Athena shows her wrapped in a blanket, with her hand over her face.
In a social media post, her father Edoardo said: “She is tiny and absolutely perfect” and the family was “completely besotted with her”.
The new baby, Athena, is the grandchild of Prince Andrew, the Duke of York, and Sarah, the Duchess of York. She becomes 11th in line to the throne.
“She is already so adored and I’m incredibly blessed to be a granny once again. So proud of Edo, Beatrice and the rest of my little 5-a-side team,” posted the duchess on social media.
The statement from Buckingham Palace gave the baby’s full name as Athena Elizabeth Rose Mapelli Mozzi, born last Wednesday at 12.57pm.
Beatrice, aged 36, and her daughter are both doing well, said the Palace, and the family are spending time with Athena’s older siblings Wolfie – Mr Mapelli Mozzi’s son from a previous relationship – and Sienna.
The parents thanked staff at the London hospital for their “wonderful care”, the same hospital where their older daughter Sienna had been born in 2021.
In ancient Greece, Athena was the goddess of war, handicraft and wisdom.