He got a BAFTA and an Oscar nomination, and went on to play the same romantic, foppish dope in quite a few other films.
A year later, just as he was becoming used to adulation, he was arrested for lewd conduct in a public place after being caught with hooker Divine Brown in a car off Sunset Strip. He was fined $1,100, given two years' probation and had to go on an AIDS awareness course.
The newspapers had a field day - the fop was really a cad, and it was such an about-turn in public perception that everyone was gripped to see what would happen to him and that nice
Meanwhile poor Hugh was complaining he had become typecast, saying: "People made two assumptions: One was that I was that character - when in fact nothing could be further from the truth... and the other frustrating thing was that they thought that's all I could do."
But his movies began to tank. Filmgoers associated him with scandal, and he couldn't quite pull off the bumbling-sweet-guy act any more.
Then film bosses realised that 'nice Hugh' had been sexualised, and offered him different work. He played womanisers, cads and rotters to critical acclaim and of course, with such a famous name, studios began shelling out millions of pounds to get him in their movies.
He enjoyed a string of beautiful girlfriends, quite a few multi-million pound houses, and a cavalcade of supercars. The films have dried up of late, but then he's so rich now he doesn't need to work.
In short, public humiliation in the international Press was the best thing that ever happened to Hugh Grant.
So I am not quite sure why he has thrown himself so wholeheartedly into the row about celebrity injunctions. He has not, to my knowledge, got an injunction out. He does dislike the Press quite profoundly, although his habit of telling photographers "I hope your kids die of cancer" is a little extreme.
He says the Press has no right to intrude on the private lives of people who have simply had the good fortune to be successful. In which he is right. Why would anyone write about him purely on the basis of his acting skills?
We do have a right though to write about criminals, and that is what Hugh was when he pleaded guilty to getting his winky out in a public place. He could not have injuncted it had he tried, because it was a matter of public record. I think it is this fact - that his sex life became public property and there was nothing he could do about it - which so riles him.
Since then, he has remained in the public eye purely and simply because of that episode - earning millions as a direct result of it - and persistently dates people who are famous or otherwise newsworthy. So his sex life has remained public property to some extent, something which must infuriate him seeing as he hasn't even been caught with a hooker again.
But why is he so in favour of injunctions? Had he been able to keep his greatest shame out of the newspapers, he would have faded into obscurity 10 years ago.
Perhaps he thinks he has got where he is today purely on the strength of his acting ability. If that's the case, where is his Oscar? Oh yes, it's in the same place as my Pulitzer.
The problem with the privacy debate is this - there isn't one. There is no massive groundswell of opinion, no campaign being backed by the public. In fact seeing as traffic on Twitter went up 14 per cent when someone tweeted the names of people linked to recent injunctions, the public seem to be pretty much in favour of openness on the topic.
This 'debate' is no such thing. It's just a slanging match with people like me on one side, who have a vested interested in reporting whatever we like, and on the other a dozen or so rich people who have all been embarrassed at some point and are having a whinge about it. The only reason Hugh Grant is getting air time is that he increases audience figures because he's famous, and he's famous because he was caught with a hooker.
I think Hugh should stop whining, get some perspective and enjoy driving around in the £143,325 Ferrari California which he would not have been able to buy were it not for the Press. That's what I'd do, and Divine Brown has the same idea.
She was an abused woman whose life finally improved after their joint arrest. She is off the streets, the £1m fees from newspaper interviews paid for her children to go to private school, and she is living happily in Atlanta with a fiance and a music production company.
She said she thanks God for sending her Hugh Grant, because it changed her life; and yet again has proved that a hooker often has a lot more heart and common sense than the clients who think a human can be rented by the hour.
If Hugh Grant, rather than launching an embittered one-man crusade, stopped to think about it or talk to anyone outside a Chelsea dinner party he might feel grateful to the Press. And he should ring up every journalist who's ever written about him, good or bad, and say thank you.
And when he's done that he should put a sock in it.
Boo hoo, poor me.