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Labour is writing to No 10 to urge the immediate appointment of the judge to lead an inquiry into the scandal.
On Friday, an unnamed 63-year-old became the third man arrested as part of the police probe.
He was arrested at an address in Surrey on suspicion of corruption. Police carried out a search of the property.
On Saturday he was released on bail until October.
Meanwhile, former News of the World (NoW) editor Andy Coulson and former royal editor Clive Goodman were released on police bail after being arrested earlier on Friday.
Mr Coulson, 43, had attended Lewisham police station in south London by appointment, and was arrested on suspicion of conspiring to intercept communications and alleged corruption.
Goodman, 53, who was jailed in 2007 for phone hacking, was arrested on suspicion of corruption.
In a letter to the prime minister
on Saturday, shadow culture secretary Ivan Lewis asks for "immediate
discussions so that by the end of the day we are in a position to agree
the appointment of the judge" to head one of the independent inquiries
into the scandal.
According to BBC political correspondent Ben Geoghegan,
"Labour say their letter has been prompted by reports that millions of
e-mails had been deleted at News International in an attempt to obstruct
Scotland Yard's investigations into phone hacking."That claim has been denied by the company," our correspondent adds.
Mr Lewis writes: "In view of the fact that the News of the World is shutting down, it is a matter of great urgency that any documentary evidence, including files and emails, is preserved to enable a proper inquiry into these serious allegations to take place."
Public inquiries On Friday, David Cameron revealed details of two new inquiries relating to the scandal.
He said a judge-led inquiry would look into "why did the first police investigation fail so abysmally; what exactly was going on at the News of the World and what was going on at other newspapers".
A second inquiry would examine the ethics and culture of the press, he added.
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David Cameron: ''The buck stops right here''
Mr Coulson denies knowledge of phone hacking when he was News of the World editor from 2003-7.
The prime minister questioned the tenability of Rebekah Brooks as News International chief executive considering she was editor of the NoW at the time murdered schoolgirl Milly Dowler's phone was allegedly being hacked.
It has been revealed that Mrs Brooks is no longer heading the firm's own inquiry into the scandal.
She told News International staff in an e-mail that those carrying out the investigation would now report to Joel Klein, a US-based senior executive at the company's owner, News Corp.
Takeover questions On Friday, Mrs Brooks held a meeting with NoW staff at its headquarters in Wapping.
A source present at the talks told the BBC she had informed staff they would eventually understand why the Sunday tabloid had to close.
She also denied closing the NoW was a "cynical ploy", and apologised for the decision.
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Rebekah Brooks' address to staff on Friday at the News of the World was captured on a secret recording
The controversy has raised questions about the proposed takeover of satellite broadcaster BSkyB by Rupert Murdoch's News Corporation, ultimate owner of the NoW.
And broadcasting regulator Ofcom has now written to the chairman of the Commons culture committee highlighting the watchdog's duty to ensure that anyone holding a broadcasting licence is a "fit and proper" person to do so.
The letter says "in considering whether any licensee remains a fit and proper person to hold broadcasting licences Ofcom will consider any relevant conduct of those who manage and control such a licence".
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