In this paper, the problem of state observation with exponential error decay for nonlinear systems affine in the input is considered and an observer is proposed. For such an observer, the drift-observability property of the system (i.e,, observability for zero input), together with an assumption on the input amplitude or on the observation relative degree, are sufficient conditions for exponential convergence of the observation error. The existence of an exponential observer is correlated to the existence of a solution for an H-infinity Riccati-like inequality. Global and semiglobal convergence results are presented.