Charlotte Gainsbourg discusses her music, her father, and her work with Lars von Trier.

Actors who think they can sing are commplace. But unlike other truant thespians – Scarlett Johansson with her wretched album of Tom Waits covers, say – Charlotte Gainsbourg has what you might call a family stake in the matter.

Her father, Serge Gainsbourg, was revered in France as a great songwriter and even greater libertine. Her mother is Jane Birkin, the beautiful English actress-singer who swapped swinging London for Paris and the arduous life of being Serge’s muse. Charlotte’s memories of music in the Gainsbourg household are of her father playing records of his own songs at top volume, to the distress of those unlucky to live next door. “The police were always coming by because he was always listening too loud,” she sighs.

I meet her in Paris, where she lives with her husband and three children. The venue is a luxury hotel near the Champs-Elysées. Gainsbourg, 40, sits at a floral sofa opposite me, elegantly casual in a black jumper, grey trousers and brown boots. Her spoken English – refined, softly spoken, composed – evokes the Chelsea bohemia of her mother’s youth. But a Gallic sheen and the occasional stumble for a word are reminders that French is her first language.

Since her father’s death in 1991, when she was 19, Gainsbourg has established herself as one France’s best-known actresses, appearing in films ranging from mainstream French comedies (Ma Femme est une Actrice, made by her husband Yvan Attal) to independent US cinema (21 Grams) and controversial art-house dramas (Lars von Trier’s Antichrist).

Her music career has followed a more haphazard route. It began in 1986 when she was a teenager with an album devised by her father, Charlotte for Ever, then didn’t resume until she was in her mid-30s. This month sees the UK release of her new album, Stage Whisper. It’s the follow-up to IRM, which she made in 2009 with Beck, the tirelessly inventive American songwriter and producer.

It was under Beck’s aegis – he wrote and produced IRM – that Gainsbourg began to inch out from her parents’ musical shadows. The album was inspired by a brain haemorrhage that Gainsbourg suffered in 2007 after a waterskiing accident. Weird clanking songs imitated the noises of the magnetic resonance imaging machines that scanned her. Slow-moving numbers mimicked an opiated state of convalescence.

Gainsbourg isn’t a technically gifted singer – her sweet, breathy voice is a one-note reproduction of her mother’s – but in Beck’s hands her vocals acquired an enigmatic pull, floating above his lushly orchestrated arrangements. Stage Whisper ties up IRM’s loose ends. It consists of a batch of new songs, several made with Beck during IRM’s recording sessions, and a CD and DVD of live performances from when she was touring the album in 2010. The tour came at the same time as she was making Lars von Trier’s film Melancholia, so she found herself shuttling from “crazy” rock festivals in Europe to the “silent, austere” shoot in Sweden.

“You need to build yourself up to go on stage, to feel you are worth it, but then to go to a film set my ego had to diminish all at once. It’s a funny exercise,” she says. It doesn’t sound very pleasant, I suggest. “No, it’s not pleasant, but it’s good. You juggle with who you are.” Was it odd playing songs about her brain injury on stage night after night? “On the contrary, it was very liberating,” she replies. “I love the sounds of the MRIs. The whole trauma was very exciting in a way.” The haemorrhage was successfully treated by surgery but surely it left her with a sense of the precariousness of life? “No, not at all.”

Gainsbourg, I learn, is prone to breezy remarks such as these. Some sound actressy (“I don’t feel that I have an identity,” she tells me at one point, which is either evidence of a radically decentred consciousness or complete tosh). Others, more sympathetically, suggest a person­ality disposed to put a positive gloss on life’s upheavals – a useful attribute, one suspects, for upbringing in the tempestuous Serge Gainsbourg/Jane Birkin household.

She began singing at 13, making her debut with her father in the duet “Lemon Incest” in which she disturbingly reprised the erotic role her mother played in the Serge-penned 1969 hit “Je t’aime ... moi non plus”. The song not surprisingly caused consternation when it came out in 1984.