Whilst pursuing her general studies, Claude Giron began learning the cello and piano at the music school in Rochefort-sur-Mer. After obtaining her baccalaureate in 1986, she enrolled in 1988 at the Paris National Conservatoire of Music and Dance (CNSMDP) in the classes of Roland Pidoux (cello) and Michel Strauss (chamber music). Three years later, she was awarded a First Prize in both disciplines.

During masterclasses held at the CNSMDP, she met Aldo Parisot. After spending a summer in Canada in his class at the Banff Academy, she decided to go and study at Yale University, where he was teaching. With a Fulbright scholarship and a grant from the Ministry of Culture, she spent three years there and obtained a Master of Music and an Artist Diploma. This period provided her with valuable experience in the field of chamber music, as she had the opportunity to work with the Tokyo String Quartet and to win several international competitions in the United States.

On her return in 1994, she joined the Radio France Philharmonic Orchestra, and then, in 1999, the Orchestre de Paris.

As well as performing chamber music in various ensembles, Claude Giron has a keen interest in all styles of music. She has recorded an album with the great jazz pianist Ahmad Jamal, and has performed in numerous concerts with chamber orchestras (European Camerata) and cello ensembles (Octuor de Paris). 
Since 1999, she has also been a member of the group Pink Martini, joining them on their European tours.
In 2006, she joined the Yiddish music group Sirba Octet, with whom she has already recorded several albums, including their latest, *Tantz*, which was a great success (Choc Classica, 3f Télérama).