This article introduces a method for performing fully Bayesian inference for nonlinear conditional autoregressive continuous-time models, based on a finite skeleton of observations. Our approach uses Markov chain Monte Carlo and involves imputing data from times at which observations are not made. It uses a reparameterization technique for the missing data, and because of the non-Markovian nature of the models, it is necessary to adopt an overlapping blocks scheme for sequentially updating segments of missing data. We illustrate the methodology using both simulated data and a data set from the S & P 500 index.