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This paper concentrates on the numerical solutions of the weakly singular Volterra integral equations with vanishing delay by means of the Nystr\"{o}m method based on the modified rectangular quadrature formula. An extrapolation type arithmetic can be constructed according to the Euler-Maclaurin expansion of error. The algorithm is beneficial to increase the convergence rate of the approach and reaches higher accuracy than the former procedures. Moreover, a posterior error estimate is derived, which can be used to formulate self-adaptive algorithm. Numerical examples demonstrate that this method is effective and applicable.