"We're very excited about the new addition to our family," Mr O'Neill said after the birth.
Princess Madeleine is the youngest child of King Carl XVI and Queen Silvia.
However, Madeleine's newborn baby is unlikely to ever ascend to the throne, as there are already nine in line before her.
These include Princess Victoria and her two children; Prince Carl Philip and his two sons; and the baby's older siblings Nicolas, aged 2, and Leonore, aged 4.