The heir to Denmark's throne, Crown Prince Frederik, and his wife, Australian-born Crown Princess Mary, are expecting their first child.
Princess Mary will continue with her royal duties for now
The pair met in a bar in Sydney during the 2000 Olympics and married in May.
The baby is due in October, and Danish media has already begun to speculate whether it will be a boy or a girl.
The constitution does not allow for female succession - although in 1953 an exception was made for the current queen, Margrethe, who had no brothers.
Danish Equal Rights Minister Eva Kjer Hansen told Danish media she felt the constitution needed to be changed.
"It would not be fair for a daughter to have to give up first place if the crown prince couple later has a son," she said.
Court officials say the princess is in good health and the pregnancy will not immediately affect her duties.
Mary, 33, is the first Australian-born princess in a European reigning family. She comes from the island of Tasmania.
Her father is a maths professor and her stepmother is British author Susan Moody. Her natural mother died in 1997.
To marry the 35-year-old crown prince, she has had to give up her Australian and British citizenship and convert to the Lutheran Church.