Abstract: In this paper, the author first gives a kind quadrature rule $\Phi_m^{*}(wf,x)$ for Cauchy type singular integral $\Phi(wf,x)$, then proves the sequence $\{\Phi_m^{*}(wf,x)\}_{m=2}^{\infty}$ is uniformly convergent to $\Phi(wf,x)$ on the interval $[-1,1]$, at that time gives its error bounds.

Key words: Cauchy type singular integral, Dini conditions, Generalized Jacobi weight.