  This paper introduces a Monte Carlo method for maximum likelihood inference
in the context of discretely observed diffusion processes. The method gives
unbiased and a.s.\@ continuous estimators of the likelihood function for a
family of diffusion models and its performance in numerical examples is
computationally efficient. It uses a recently developed technique for the exact
simulation of diffusions, and involves no discretization error. We show that,
under regularity conditions, the Monte Carlo MLE converges a.s. to the true
MLE. For datasize $n\to\infty$, we show that the number of Monte Carlo
iterations should be tuned as $\mathcal{O}(n^{1/2})$ and we demonstrate the
consistency properties of the Monte Carlo MLE as an estimator of the true
parameter value.
