The actor, who sits on the board of the Hacked Off campaign group, has 'reluctantly' joined Twitter as @hackedoffhugh ahead of the Leveson report publication tomorrow.
He has not yet tweeted, but already has more than 4,400 followers. The blurb on his account - which has been verified by Hacked Off - says: "On the board of Hacked Off, and hacked off enough to tweet."
Grant, who has been the subject of several negative newspaper stories in the past, has repeatedly called for a state-appointed system of regulation for the print media.
The industry is currently self-regulated via the Press Complaints Commission, but the body has been widely criticised as being ineffective.
Lord Justice Leveson was asked to investigate standards in the print media in July 2011 after the phone hacking scandal at the News of the World.
Ahead of the report's publication tomorrow, a group of MPs and peers has warned that state regulation would endanger free speech.
But Grant told ITV1's This Morning that the report is "a chance in a generation and it really cannot be allowed to slip".
Speaking alongside radio broadcaster Nick Ferrari, Grant also expressed his hope that David Cameron will "dare" to impose state regulation if Leveson does, as expected, recommend it in the report.
"I thought Leveson seemed to me to be conducting a very balanced and fair enquiry and he seemed like a decent man, so God knows what he'll say," said Grant, who gave evidence before the inquiry last year.
"But I think if he does recommend that finally the press can no longer be allowed to mark their own homework - if he does say that, then it will be down to the prime minister really to do something that no prime minister has dared to do in the last 50 years in the same situation."
Downing Street said it has today received around "half a dozen copies" of the Leveson report and, alongside Cameron, it is thought that deputy prime minister Nick Clegg and culture secretary Maria Miller will also review the document.
Speaking at Prime Minister's Questions, Cameron said that he was "looking forward to reading the report carefully", and noted that the status quo in press regulation is "unacceptable and needs to change".