This article presents a new simulation-based technique for estimating the likelihood of stochastic differential equations. This technique is based on a result of Dacunha-Castelle and Florens-Zmirou. These authors proved that the transition densities of a nonlinear diffusion process with a constant diffusion coefficient can be written in a closed form involving a stochastic integral. We show that this stochastic integral can be easily estimated through simulations and we prove a convergence result. This simulator for the transition density is used to obtain the simulated maximum likelihood (SML) estimator. We show through some Monte Carlo experiments that our technique is highly computationally efficient and the SML estimator converges rapidly to the maximum likelihood estimator.