In this paper, we develop a new numerical method with asymptotic stability properties for solving stochastic differential equations (SDEs). The foundations for the new solver are the Steklov mean and an exact discretization for the deterministic version of the SDEs. Strong consistency and convergence properties are demonstrated for the proposed method. More- over, a rigorous linear and nonlinear asymptotic stability analysis is carried out for the mul- tiplicative case in a mean-square sense and for the additive case in a path-wise sense using the pullback limit. In order to emphasize the characteristics of the Steklov discretization we use as benchmarks the stochastic logistic equation and the Langevin equation with a nonlinear potential of the Brownian dynamics. We show that the Steklov method has mild stability requirements and allows long-time simulations in several applications.