In this paper, we propose a new observer design for nonlinear systems that can be linearized using a change of coordinates and a singular time re-scaling. Our observer is a switched system and the observer error dynamics are described, after time re-scaling, by a switched linear system that is uniformly exponentially stable. We also give necessary and sufficient conditions for linearizability via a change of coordinates and a singular time re-scaling. Our methods are illustrated by an example coming from the ABS literature.