Eight Cabinet ministers were named as “core participants” – people who have a significant interest in the hearings or may face criticism.
The move came ahead of potentially explosive testimony next week from former News International executives Rebekah Brooks and Andy Coulson.
Lord Justice Leveson also gave core participant status to Deputy Prime Minister Nick Clegg, Business Secretary Vince Cable, Education Secretary Michael Gove, Justice Secretary Kenneth Clarke, Home Secretary Theresa May and Chancellor George Osborne.
All of the ministers, apart from Mr Osborne, will give evidence in person over the coming weeks.
The core participant application was initially made on behalf of the government. But Lord Justice Leveson rejected this on legal grounds, observing that under Britain’s unwritten constitution, the government has “no independent existence in law”.
He agreed that the individual ministers could seek the status, noting that they are not seeking the right of core participants to question other witnesses or make opening and closing submissions.
James Eadie, QC, representing the government, said the ministers had a “clear public interest” in the inquiry’s proceedings because they will be responsible for acting on Lord Justice Leveson’s recommendations.
He said: “The important thing is that those who need to have advance sight of relevant material, have advance sight of relevant material.”
All but one of the ministers have already submitted written witness statements to the inquiry, the hearing was told.
Lord Justice Leveson stressed: “There can be no question of access being sought for the purpose of preparing evidence.”
The inquiry chairman insisted that all the ministers and their advisers must sign a confidentiality pledge not to leak any of the material they see in advance.
He said: “I mean absolutely no discourtesy to ministers, or those who will have to assist them, but the rule must apply to everyone.”
Without naming him, Lord Justice Leveson accused Labour MP Chris Bryant, who has core participant status, of showing “total disregard” for the confidentiality agreement he has signed. Speaking in the House of Commons last week, Mr Bryant quoted from a then-unreleased document listing meetings between Mr Cameron and media mogul Rupert Murdoch.
Existing core participants for the inquiry’s third module, which is looking at relations between the press and politicians, include Mrs Brooks, national newspaper groups and a number of current and former MPs.
The government faced calls for Mr Hunt’s resignation last week after the Leveson Inquiry released e-mails detailing contacts between his office and a senior executive at Rupert Murdoch’s News Corporation.
Labour claimed the messages showed that the Culture Secretary failed to fulfil his quasi-judicial role in relation to Mr Murdoch’s proposed takeover of satellite broadcaster BSkyB.
Mr Hunt’s special adviser, Adam Smith, quit the next day, admitting he went “too far” in his contacts with News Corp lobbyist Frederic Michel.
If the ministers had been core participants at the time, they would have had advance access to the e-mails and could have argued that parts should be blanked out before being made public.