  We characterise smooth curves in a smooth cubic threefold whose blow-ups
produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree
$d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a
3-secant line in the cubic threefold. Among the list of ten possible such types
$(g,d)$, two were previously left as open numerical possibilities, namely
$(g,d) = (0,5)$ and $(2,6)$. Using the Sarkisov link associated with a curve of
type $(2,6)$, we are able to produce the first example of a pseudo-automorphism
with dynamical degree greater than $1$ on a smooth threefold with Picard number
$3$. We also prove that the group of birational selfmaps of any smooth cubic
threefold contains elements contracting surfaces birational to any given ruled
surface.
