  We consider a self-adjoint two-dimensional Schr\"odinger operator
$H_{\alpha\mu}$, which corresponds to the formal differential expression \[
-\Delta - \alpha\mu, \] where $\mu$ is a finite compactly supported positive
Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $\alpha
>0$ is the coupling constant. It was proven earlier that $\sigma_{\rm
ess}(H_{\alpha\mu}) = [0,+\infty)$. We show that for sufficiently small
$\alpha$ the condition $\sharp\sigma_{\rm d}(H_{\alpha\mu}) = 1$ holds and that
the corresponding unique eigenvalue has the asymptotic expansion $$
  \lambda(\alpha) = -(C_\mu + o(1))\exp\Big(-\tfrac{4\pi}{\alpha\mu({\mathbb
R}^2)}\Big), \qquad \alpha\rightarrow 0+, $$ with a certain constant $C_\mu >
0$. We obtain also the formula for the computation of $C_\mu$. The asymptotic
expansion of the corresponding eigenfunction is provided. The statements of
this paper extend Simon's results, see \cite{Si76}, to the case of
potentials-measures. Also for regular potentials our results are partially new.
