Let $\mathfrak G_m$ be the class of indecomposable probability laws with bounded spectrum $S(G)$ where \begin{gather}
m=\min(u,v),\ u=G({\inf S(G)}),\ v=G({\sup S(G)}),\
G({x})=G(x+0)-G(x).
\end{gather}If$G_1\ast G_2\in\mathfrak G_m$,$m>0$,$F_1$has median 0 and if the uniform metric$\rho(F_1\ast F_2,G_1\ast G_2)\le\varepsilon$then there exists a constant$\varepsilon_0=\varepsilon_0(G)>0$such that$$ \min\{\rho(F_1,G_1),\rho(F_1,G_2)\}\le(m-\sqrt{m^2-4\varepsilon})/2 $$when$0\le\varepsilon\le\varepsilon_0$, and this estimate cannot be improved in the class$\mathfrak G_m$.