Sign languages (SL) require a fundamental rethinking of many basic assumptions about human language processing because instead of using linear speech, sign languages coarticulate facial expressions, shoulder and hand movements, eye gaze and usage of a three-dimensional space. SL researchers have therefore advocated SL-specific approaches that do not start from the biases of models that were originally developed for vocal languages. Unfortunately, there are currently no processing models that adequately achieve both language comprehension and formulation, and the SL-specific developments run the risk of becoming alienated from other linguistic research. This paper explores the hypothesis that a construction grammar architecture offers a solution to these problems because constructions are able to simultaneously access and manipulate information coming from many different sources. This claim is illustrated by a proof-of-concept implementation of a basic grammar for French Sign Language in Fluid Construction Grammar.