  For all $k\in]0,1[$, we construct a canonical bijection between the space of
ramified coverings of the sphere and the space of complete immersed surfaces in
$3$-dimensional hyperbolic space of finite area and of constant extrinsic
curvature equal to $k$. We show, furthermore, that this bijection restricts to
a homeomorphism over each stratum of the space of ramified coverings of the
sphere.
