  We define and study the renormalized volume for geometrically finite
hyperbolic $3$-manifolds, including with rank-$1$ cusps. We prove a variation
formula, and show that for certain families of convex co-compact hyperbolic
metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric $g_0$
with rank-$1$ cusps, the renormalized volume converges to the renormalized
volume of the limiting metric.
