The names were bought up two days after Carys was born

Douglas, 58, took the precaution just two days after Carys Zeta Douglas was born back on 20 April in New Jersey, according to UK newspapers on Thursday.

He is thought to want to protect her name from so-called "cyber-squatters", who can make money by buying potentially lucrative domain names for themselves.

Although Douglas is unlikely to have paid more than a few dollars, the addresses could be worth a lot of money if Carys grows up to be as famous as her parents.

Carys' internet names are registered under her father's name in London and Melbourne with a PO Box address in Los Angeles.

When Tony Blair's son Leo Blair was born, there was a sudden rush for LeoBlair.com on the basis that it could be a money spinner Paul Coulson, Simply.com

They are also registered through a property company in London which has bought catherinezetajones.com, and catherinezetajones.biz

According to Paul Coulson, managing director of the domain name company Simply.com, it is "absolutely essential" for a celebrity to register his or her name.

"When Tony Blair's son Leo Blair was born, there was a sudden rush for LeoBlair.com on the basis that it could be a money spinner." he said.

"Those who bought it could then sell it for money."

But Mr Coulson said this was less prevalent nowadays as celebrities are on top of the game and what tends to happen now is 'cyber-squatters' registering well known names but mis-spelling them.

"If an internet user types a name in and mis-spells it, they then go to another site." he said.

"That site could be selling something, or could even be a porn site."

"If you're starting a singing or acting career, or are in the public eye like Carys, you need to register your domain name so it isn't used improperly."