The context of medical conditions is an important feature to consider when processing clinical narratives. NegEx and its extension ConText became the most well-known rule-based systems that allow determining whether a medical condition is negated, historical or experienced by someone other than the patient in English clinical text. In this paper, we present a French adaptation and enrichment of FastContext which is the most recent, n-trie engine-based implementation of the ConText algorithm. We compiled an extensive list of French lexical cues by automatic and manual translation and enrichment. To evaluate French FastContext, we manually annotated the context of medical conditions present in two types of clinical narratives: (i) death certificates and (ii) electronic health records. Results show good performance across different context values on both types of clinical notes (on average 0.93 and 0.86 F1, respectively). Furthermore, French FastContext outperforms previously reported French systems for negation detection when compared on the same datasets and it is the first implementation of contextual temporality and experiencer identification reported for French. Finally, French FastContext has been implemented within the SIFR Annotator: a publicly accessible Web service to annotate French biomedical text data (http://bioportal.lirmm.fr/annotator). To our knowledge, this is the first implementation of a Web-based ConText-like system in a publicly accessible platform allowing non-natural-language-processing experts to both annotate and contextualize medical conditions in clinical notes.