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In this paper we consider the Liouville equation $\Delta u +\lambda^2e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doublyconnected 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 sequenceof 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$\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega e^{\,u_n}\,dx\to 8\pic_\Omega$, where $c_\Omega$ is a constant depending on the conformal class of$\Omega$ only.

Maximal solution of the Liouville equation in doubly connected domains