What is it for a predicate or a general term to be a rigid designator? Two strategies for answering this question can be found in the literature, but both run into severe difficulties. In this paper, it is suggested that proper names and the usual examples of rigid predicates share a semantic feature which does the theoretical work usually attributed to rigidity. This feature cannot be equated with rigidity, but in the case of singular terms this feature entails their rigidity, as understood in the standard characterisation. Hence, it is appropriate to call this feature proto-rigidity.