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This study investigates the problem of delay-dependent robust stability for uncertain stochastic systems with interval time-varying delay. No restrictions on the derivative of the time-varying delay are imposed, although lower and upper bounds of the delay interval are assumed to be known. By decomposing the delay interval into multiple equidistant subintervals, new Lyapunov-Krasovskii functionals are introduced on these intervals, which result in being much less conservative than most existing results in the literature. In addition, the reduction in the conservatism of the proposed stability criteria is also attributed to the free-weighting matrix technique and a newly proposed bounding condition, which is used to estimate the upper bound of stochastic differential of the Lyapunov-Krasovskii functional without neglecting any useful terms. All the stability criteria are formulated in the form of linear matrix inequalities (LMIs), which are dependent on the size of the time delay. Some numerical examples are given to show the less conservatism and applicability of the obtained results. Moreover, when the number of the divided subintervals increases, the corresponding criteria can provide an improvement on the results.