  We consider the problem of prescribing Gaussian and geodesic curvatures for a
conformal metric on the unit disk. This is equivalent to solving the following
P.D.E. \begin{equation*}\begin{cases}-\Delta
u=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_\nu u+2=2h(z)e^\frac
u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are
the prescribed curvatures. We construct a family of conformal metrics with
curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as
$\varepsilon$ goes to $0$, which blows up at one boundary point under some
generic assumptions.
