Jude Law Says He's 'Proud' of How He Dealt with His Early Fame: 'I Handled It Pretty Well' (Exclusive)

In 1998, Jude Law, who is currently starring in the new film Firebrand, had a thriving acting career — but he was still living in relative obscurity, as far as international fame goes.

Born and raised in London, the Brit had had several successful turns in theater, and had also been a steady character on the BBC TV show Shopping where he met his future wife, Sadie Frost.

But when he starred in The Talented Mr. Ripley in 1999, alongside Matt Damon and Gwyneth Paltrow, that's when he says everything changed.

"It was just a wonderful period of my life," Law, now 51, tells PEOPLE of becoming a bona fide Hollywood star. "It felt like everything I'd hoped acting and that career could provide was happening."

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At first, it was nothing but fabulous.

"I was working with a wonderful director, a really exciting young cast," he says. "I was playing a guy who was just playing jazz and boating around and swimming in the ocean and eating delicious food in wonderful locations."

The film was a huge success, and suddenly, Law says that overnight, he was famous famous: "The success, or the response it got, was life-changing. Really."

It wasn't just that he was suddenly invited to high-profile events, like the Golden Globes and Academy Awards, getting tons of offers to work with the best directors. He was. But he also says his private life was suddenly a hot topic.

"I think there was an expectation suddenly and attention to choices I was making, like who was I working with?" the Closer actor tells PEOPLE. "And then ultimately what was going on in my private life was scrutinized too. And all of that was quite a lot. I was so young."

"I look back now — my eldest son is 27, and I wasn't much older," he says of his early days of fame. "And honestly, I look back and I'm really proud. I think I handled it pretty well."

For more from Jude Law, pick up the latest issue of PEOPLE, on newsstands Friday.