  For a compact surface $X_0$, Thurston introduced a compactification of its
Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary
$\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We
introduce a similar bordification for the Teichm\"uller space $\mathcal T(X_0)$
of a noncompact Riemann surface $X_0$, using the technical tool of geodesic
currents. The lack of compactness requires the introduction of certain
uniformity conditions which were unnecessary for compact surfaces. A technical
step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$,
may be of independent interest.
