The gating mechanism of a single–ion channel is usually modelled by a finite–state–space continuous–time Markov chain. The patch–clamp technique enables the experimenter to record the current flowing across a single–ion channel. In practice, the current is corrupted by noise and low–pass filtering, and is sampled with a typically very short sampling interval. We present a method for performing Bayesian inference about parameters governing the underlying single–channel gating mechanism and the recording process, directly from such single–channel recordings. Our procedure uses a technique known as Markov chain Monte Carlo, which involves constructing a Markov chain whose equilibrium distribution is given by the posterior distribution of the unknown parameters given the observed data. Simulation of that Markov chain then enables the investigator to estimate the required posterior distribution. As well as providing a method of estimating the transition rates of the underlying Markov chain used to model the single–channel gating mechanism and the means and variances of open and closed conductance levels, the output from our Markov chain Monte Carlo simulations can also be used to estimate single–channel properties, such as the mean lengths of open and closed sojourn times, and to reconstruct the unobserved quantal signal which indicates whether the channel is open or closed. The theory is illustrated by several numerical examples taken mainly from the ion–channel literature.