We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times i *delta, i = 1 ... N. Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we prove limit theorems for functionals associated with the observations, based on local means of the sample. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent. We provide an illustration on simulated and real data from neuronal activity.