Nope, I don't know what the hell was Leveson but I am against government intervention in the press, in the UK or in Australia or in Madagascar.
I don't trust the government to define the criteria for journalistic integrity. There are courts for anyone who feels wronged go to. Regulating content is censorship.
(Leveson's the ongoing UK inquiry into the hacking and police bribery at Newscorp.)
The problem we have currently is that the Australian Press Council seems to be very weak and therefore not doing its job.
Whether the solution is a statutory body, I'm a bit skeptical, but a problem does exist with existing codes of practice being both out of date and not very well adhered to. Personally I think giving it some funding to make it more independent of the publishers would probably be a good idea. At present half its money comes from News Limited, so of course it's going to be constrained by that. Withdrawal by either of our two large media companies would likely completely collapse the current Press Council.
(Thinking about it, I suspect the extreme concentration in Australian media is part of what makes the current self-regulation very weak. If there were more media companies providing smaller shares of the funding it would presumably be a more effective body.)
Another idea suggested in the inquiry report is to make present legislated press priveleges (exemption from privacy law, and from the provisions around misleading conduct in trade in the Consumer and Competition Act) provisional on membership of the APC peak body.
At any rate, if such a body were created to replace the APC, it's being proposed even in the sensationalist OP that it have the power to use the courts (or possibly to function as one) rather than of direct regulation. Is it government censorship to sue a news outlet for defamation or libel? Would it be censorship to sue a news outlet for breaching the code of practice it has itself signed up to?