In this article, we address the generalized BBM equation with white noise dispersion which reads du - du(xx) + ux o dW + u(p)u(x)dt = 0, in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the norm of the solutions decays to zero at as t approaches to , by assuming that and that the initial data is small in . This decay rate matches the one for solutions of the linear equation with white noise dispersion.