The author develops a modification of the method of R. Piessens and P. Verbaeten [BIT 13, 451-457 (1973; Zbl 0266.65081)] for approximating the solution \(f\) of the integral equation \(\int^ x_ 0 (t(x)-t(y))^{-\alpha}f(y)dy=g(x)\), \(0\leq x\leq 1\), \(0<\alpha<1\), where \(t\) is strictly increasing and continuously differentiable. The modification is of algorithmical nature, namely the approximate solution itself is expressed by aid of Chebyshev polynomials. In certain cases this new method requires fewer calculations. The paper is concluded by two nice illustrating examples.