Bessy trained at the Paris Opera Ballet School from the age of ten, the youngest student ever admitted, and joined the Paris Opera Ballet at age 13, the youngest danseuse ever admitted.
Bessy was closely associated with Serge Lifar and created leading roles in his 1951 Snow White, 1955 Noces fantastiques and 1958 Daphnis and Chloe.
Bessy was featured in Gene Kelly's film Invitation to the Dance (1956), and four years later he created Pas de dieux at the Paris Opera for her.
As director of the Paris Opera Ballet School, she introduced profound reforms to the teaching regime which led to the birth of a new generation of highly technical dancers like Sylvie Guillem, Patrick Dupond, Élisabeth Platel, Marie-Claude Pietragalla, and succeeded in organising the construction of a new school building in Nanterre inaugurated in 1987.
Bessy was named to France's Ordre national du Mérite (dignity of the Grand Cross, its highest class), in 2009.