  In this paper we consider the Liouville equation $\Delta u +\lambda^2
e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly
connected domain $\Omega$. We show that there exists a simple, closed curve
$\gamma\subset \Omega$ such that for a sequence $\lambda_n\to 0$ and a sequence
of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{\lambda_n}}\to H$,
where $H$ is a harmonic function in $\Omega\setminus\gamma$ and
$\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega e^{\,u_n}\,dx\to 8\pi
c_\Omega$, where $c_\Omega$ is a constant depending on the conformal class of
$\Omega$ only.
