Useful references for your question are Robert Brooks' "Platonic surfaces" and Dan Mangoubi's "Conformal Extension of Metrics of Negative Curvature" (both on arxiv).

I emailed Luca Migliorini requesting his paper. He told me it was basically his undergraduate thesis, published in a defunct italian journal, and that no copy of it remains. In otherwords, utterly useless.

The basic fact on compactifying a riemann surface is this: if $S$ is a finite area riemann surface, then there exists a compact riemann surface $S^c$ and a finite set of points $p_1, \ldots, p_k$ on $S^c$ such that $S^c \setminus {{p_1, \ldots, p_k}}$ is conformally equivalent to $S$.

In Brooks' paper, he states that this riemann surface $S^c$ is unique. However I'll admit to not be convinced of this uniqueness. The expression he uses throughout is "conformally filling punctures" -- a phrase which I think deserves more explanation than is given.

Lemma 1.1 in Brooks is interesting, and justifies the above claim. Of course we know what cusps on riemann surfaces look like. A cuspidal neighborhood $C$ of a Riemann surface can be taken isometric to the quotient of $\{ z\in \mathbb{H}^2: \Im(z)\geq 1/y \}$ by the isometry $z\mapsto z+1$, for some $y>0$. The parameter $y$ gives a measure on the size of the cusp, i.e. gives a geodesic loop homotopic to the puncture with hyperbolic length $y$. So the cusp $C$ is really isometric to the punctured ball of euclidean radius proportional to $1/y$ via the mapping $z\mapsto e^{2\pi i z}$ on the punctured open unit disk $D^\ast$ equipped with the metric $ds^*=\frac{-1}{r \log r} |dz|$. However $ds^*$ blows-up as $r\to 0$ like $1/r$.

Brooks (and afterwords Mangoubi more explicitly) gives, for any $\epsilon>0$, smooth bump functions $\delta$ concentrated at the origin on $D$ such that $e^\delta ds^*$ extends to a smooth metric past the origin and whose curvature remains pinched $-1 \pm \epsilon$.

I am going to include the details of this construction, together with some remarks relating to Donaldson's compactification of algebraic curves (from his book) shortly.