  We study ground states of two-dimensional Bose-Einstein condensates with
attractive interactions in a trap $V(x)$ rotating at the velocity $\Omega $. It
is known that there exist a critical rotational velocity $0<\Omega
^*:=\Omega^*(V)\leq \infty$ and a critical number $0<a^*<\infty$ such that for
any rotational velocity $0\le \Omega <\Omega ^*$, ground states exist if and
only if the coupling constant $a$ satisfies $a<a^*$. For a general class of
traps $V(x)$, which may not be symmetric, we prove in this paper that up to a
constant phase, there exists a unique ground state as $a\nearrow a^*$, where
$\Omega\in(0,\Omega^*)$ is fixed. This result extends essentially our recent
uniqueness result, where only the radially symmetric traps $V(x)$ could be
handled with.
