A multirate sampled-data observer for nonlinear systems under asynchronous sampling is developed in this paper. The proposed multirate observer is based on a continuous-time design coupled with intersample output predictors for the sampled measurements. The sampled-data system with the multirate observer forms a hybrid system and it is shown that the error dynamics of the overall system is input-to-output stable with respect to measurement errors, by applying the Karafyllis-Jiang vector small-gain theorem. This sampled-data design also offers robustness with respect to perturbations in the sampling schedule. The proposed method is evaluated through linear systems and a nonlinear batch reactor example.