The picture shows Prince Alexander sleeping soundly on some crisp white linen.
Carl Philip, 36, is the second child and only son of King Carl XVI Gustaf and Queen Silvia of Sweden. His elder sister Princess Victoria is first in line to the throne.
There had been rumours ahead of Thursday's press conference that their son would be named Duke of Dalarna, the region where Sofia grew up.
Alexander already has plenty of royal cousins to play with. Prince Oscar, the second child of Princess Victoria, was born just seven weeks ago. He is the younger brother of four-year-old Princess Estelle.
London-based Princess Madeleine also has two children, Princess Leonore and Prince Nicolas.