  In the study of immersed surfaces of constant positive extrinsic curvature in
space-forms, it is natural to substitute completeness for a weaker property,
which we here call quasicompleteness. We determine the global geometry of such
surfaces under the hypotheses of quasicompleteness. In particular, we show
that, for $k>\text{Max}(0,-c)$, the only quasicomplete immersed surfaces of
constant extrinsic curvature equal to $k$ in the $3$-dimensional space-form of
constant sectional curvature equal to $c$ are the geodesic spheres. Together
with earlier work of the author, this completes the classification of
quasicomplete immersed surfaces of constant positive extrinsic curvature in
$3$-dimensional space-forms.
