Statelessness in France is the subject of a specific procedure before the French Office for the Protection of Refugees and Stateless Persons (OFPRA).
If you wish to apply for statelessness status, you must contact OFPRA by post to request a statelessness application form. This request must be written in French, include the surname(s), first name(s), and address of the applicant, as well as the specific reasons for his/her request.
If you are recognized as stateless, you are entitled to a multi-year residence permit for a maximum of 4 years, renewable, and to a travel document.
If you are recognized as stateless, you will find more information in the Welcome booklet for people recognized as stateless (in french).