The authors give a new simplified proof that the infinite Bernoulli convolution \(\nu_\lambda\) of the measures \[ \textstyle{{1\over 2}} (\delta_{-\lambda^n}+\delta_{\lambda^n}) \] for \(n\in\mathbb{N}\) and \({1\over 2}<\lambda<1\) is absolutely continuous with \(L^2\)-density. The proof is based on differentiation techniques for measures.