A candid conversation with French singer Lou Doillon

Lou Doillon is the daughter of French film royalty as well as a well-known actress and model in France, but it's a love of music that's making her name these days. Until five years ago, she kept her songs to herself.

"Music came as a kind of secret garden to protect myself against -- against or with the rest of the exposure, so it was extremely private," Doillon told "CBS This Morning: Saturday" co-host Anthony Mason. "The best part of it I guess is that I thought I was a born groupie. So I was always around musicians... And the boyfriends seem to have a common link of all being guitarists and singers."

"That's very dangerous," Mason said, smiling.

"Yeah! Well, it took me a long time to think that actually I was in love with the guitar maybe more than them," Doillon said, laughing.

The daughter of French film director Jacques Doillon, she was just 6 when she acted in her first film with her British mother, Jane Birkin, who'd already gained fame as the muse of legendary singer Serge Gainsbourg.

Her older half-sister, Charlotte Gainsbourg, also became a renowned actor and singer. But while Doillon emerged as one of French fashion's "it" girls, she struggled to find her own place in the shadow of her family's notoriety.

"I was racing to compete or to just do something to love myself enough to not hate all those people around me because I respected them so much. ... And I suddenly surrendered when I was 27, 28, thinking, you know, never mind, I'm never gonna make it," Doillon said.

"What did 'not making it' mean? 'Cause a lot of people would look at what you'd achieved at that point and say, 'How could you say you haven't made it?'" Mason asked.

"Well, I guess that it was, you know, doing nearly 20 movies that had never really had the success I had hoped they would have," Doillon said.

But then a friend reminded her of the songs she'd always written in her journals.

"He was very sweet 'cause he said, 'The best things about you, you have hidden.' And I thought 'Well, because it's what I really am, and I haven't gotten the guts to actually show that. And 'cause people are going to hate me,'" Doillon said. "And he was brilliant. He said, 'They already do. So what have you got to lose?' And I thought, 'OK.'"

Her debut album, "Places," released in 2012, went double platinum in France. Nominated for Best Female Artist at the Victoires de la Musiques awards, the French Grammys, Doillon was convinced she had no chance to win.

"I don't even remember hearing my name. ... I could feel that there were many movements of cameras suddenly running towards me, and then I thought, 'Run, take it, because even if they got it wrong, they wouldn't dare take it away,'" Doillon recounted.

The 33-year-old singer still isn't sure the past few years have been entirely real.

"It was like a fairy tale," Doillon said.

"So do you believe in it now?" Mason asked, as Doillon laughed.

"It's been such pleasure, it's like -- yeah -- tasting the best food on the planet," Doillon said. "The idea that someone could take it away is frightening, but thank God --"

"But who's gonna take it away now?" Mason asked.

"Yeah, that's what I think," Doillon said. "That's what I -- when I have a massive panic falling asleep, I do believe that if there are still five people willing to come in my kitchen, I can still sing them songs. And no one can take that away."