The paper studies the polynomial convergence of solutions of a scalar nonlinear Itˆo stochastic differential equation dX(t) = −f(X(t)) dt + (t) dB(t) where it is known, a priori, that limt!1 X(t) = 0, a.s. The intensity of the stochastic perturbation is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function f obeys limx!0 sgn(x)f(x)/|x| = a, for some > 1, and a > 0. We study two asymptotic regimes: when tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when 0). When decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.