An exact maximum likelihood method is developed for the estimation of parameters in a non-Gaussian nonlinear log-density function that depends on a latent Gaussian dynamic process with long-memory properties. Our method relies on the method of importance sampling and on a linear Gaussian approximating model from which the latent process can be simulated. Given the presence of a latent long-memory process, we require a modification of the importance sampling technique. In particular, the long-memory process needs to be approximated by a finite dynamic linear process. Two possible approximations are discussed and are compared with each other. We show that an auto-regression obtained from minimizing mean squared prediction errors leads to an effective and feasible method. In our empirical study we analyze ten log-return series from the S&P 500 stock index by univariate and multivariate long-memory stochastic volatility models.