  The first measurements of CP violation in the $B$ system will likely probe
$\sin 2\alpha$, $\sin 2\beta$ and $\cos 2\gamma$. Assuming that the CP angles
$\alpha$, $\beta$ and $\gamma$ are the interior angles of the unitarity
triangle, these measurements determine the angle set $(\alpha,\beta,\gamma)$
except for a twofold discrete ambiguity. If one allows for the possibility of
new physics, the presence of this discrete ambiguity can make its discovery
difficult: if only one of the two candidate solutions is consistent with
constraints from other measurements in the $B$ and $K$ systems, one is not sure
whether new physics is present or not. We review the methods used to resolve
the discrete ambiguity and show that, even in the presence of new physics, they
can usually be used to uncover this new physics. There are some exceptions,
which we describe in detail. We systematically scan the parameter space and
present examples of values of $(\alpha,\beta,\gamma)$ and the new-physics
parameters which correspond to all possibilities. Finally, we show that if one
relaxes the assumption that the bag parameters $\BBd$ and $\BK$ are positive,
one can no longer definitively establish the presence of new physics.
