The behaviour of a standard super-twist controller under stochastic perturbations, when its dynamic is governed by the stochastic differential equation of Ito type with discontinuos right-hand side, is studied. The suggested analysis is based on the Lyapunov functions V-t = V (x(t), y(t)) (Polyakov-Poznyak, Moreno-Osorio, Orlov and Utkin) designed for the stability analysis of the deterministic version of super-twist controllers. The major finding is that under stochastic (in fact, unbounded) perturbations, the special selection of a gain-parameter of such controller, making it depending on V-t and its gradient partial derivative V/partial derivative y (x(t), y(t)), provides the controller with an adaptivity property and guarantees the mean-square convergence of V-t into the prespecified zone around the origin which depends on the diffusion parameter of stochastic noise, the upper estimate of the second derivative partial derivative V-2/partial derivative y(2) as well as on the parameters of the controller.