  We compute the first cohomology spaces
$H^1(\mathfrak{osp}(1|2);\mathfrak{D}_{\lambda,\mu})$ ($\lambda,
\mu\in\mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with
coefficients in the superspace $\mathfrak{D}_{\lambda,\mu}$ of linear
differential operators acting on weighted densities on the supercircle
$S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In
fact, we prove here that the situation is a little bit more complicated. (To
appear in LMP.)
