Goodman was suspended by the News of the World after his arrest
Clive Goodman, 48, from London, was arrested after claims by the Prince of Wales's household of security breaches.
Glenn Mulcaire, 35, admitted the same charge and five counts of intercepting messages on the mobile phones of the likes of publicist Max Clifford.
Sentencing will take place after details of the case are outlined at an Old Bailey hearing in January.
News of the World editor Andy Coulson, said he apologised "unreservedly" on behalf of the newspaper to Prince William and Prince Harry, Paddy Harverson, Helen Asprey and Jamie Lowther-Pinkerton "for the distress caused by this invasion into their privacy".
In a statement, he said: "As the editor of the newspaper, I take ultimate responsibility for the conduct of my reporters.
"Clive Goodman's actions were entirely wrong and I have put in place measures to ensure that they will not be repeated by any member of my staff."
The other targets of Mulcaire's actions were said to include the Liberal Democrat MP Simon Hughes and model Elle Macpherson.
Two other victims were in the sporting world - the chairman of the Professional Footballers Association, Gordon Taylor, and England footballer Sol Campbell's agent Sky Andrew.
The case came to light in November 2005 following the publication of a story in the News of the World about Prince William's knee injury.
William began to fear aides' mobile phone voicemail messages were being intercepted.
Clive Goodman's barrister
His suspicions were raised further when an article by Mr Goodman claimed that the prince had been lent some broadcasting equipment by ITV's political editor, Tom Bradby.
Mr Bradby said that when he and William met later, "we both looked at each other and said 'Now, how on earth did that get out?'.
"... the answer we came up with is that it must be something like breaking into mobile answering machine messages."
Complaints by three staff at Clarence House sparked the police inquiry which was widened to examine whether other public figures had had calls intercepted.
Lib Dem MP Simon Hughes said that tapping or intercepting messages should only be done with the authority of the law - and in exceptional cases such as terrorism.
"Intercepting personal voicemail messages is a completely unacceptable breach of privacy - whether the victim is a royal prince, a politician, or someone completely out of the public eye.
"People who leave messages and those intended to receive them are all entitled to have private conversations.
"We live in an age where invasions of privacy are becoming more frequent. This does not make the practice any more acceptable."
He added: "I have long held the view that courts should be allowed to deprive those responsible for this sort of behaviour of their liberty because it is a serious offence to interfere with the freedoms of others.
"Some people, like me, are resilient enough to take this sort of behaviour more or less in their stride, but other people are not, and nobody should have to."
Goodman was suspended by the UK's best-selling newspaper after he was charged in August.
He was responsible for a raft of exclusives in his time and at 2002's Real Press Awards was named Royal Editor of the Year.
After the pleas were entered, Goodman's counsel John Kelsey-Fry said the journalist wanted to apologise to the members of the Royal Family concerned.
"He wishes through me to take the first opportunity to apologise publicly to those affected by his actions.
"He accepts they were a gross invasion of privacy and Mr Goodman accepts that this characterisation is correct.
"He therefore apologises unreservedly to the three members of the royal household staff concerned and their principals, Prince William, Prince Harry and the Prince of Wales."