The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity $\omega (·)$ satisfies $$\underset{s\downarrow 0}{lim\; sup}\omega \left(s\right)ln\frac{1}{s}=0$$ are considered on these boundaries. Functions satisfying the Hölder condition of order $\alpha$, $0<\alpha \le 1$, belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise...