  In the standard model the CP asymmetries in $B\to \psi K_S$ and $B\to \psi
K_L$ are equal in magnitude and opposite in sign to very good approximation. We
compute the order $\epsilon_K$ corrections to each of these CP asymmetries and
find that they give a deviation from $\sin 2\beta$ at the half percent level,
which may eventually be measurable. However, the correction to $a_{CP}(B\to
\psi K_S)+a_{CP}(B\to \psi K_L)$ due to $\epsilon_K$ is further suppressed. The
dominant corrections to this sum, at the few times $10^{-3}$ level, come from
the $B$ lifetime difference, and CP violation in $B- \bar B$ mixing and $B \to
\psi K$ decay. New physics could induce a significant difference in the
$\sin(\Delta m_B t)$ time dependence in the asymmetries if and only if the
"wrong-flavor" amplitudes $B \to \psi \bar K$ or $\bar B \to \psi K$ are
generated. A scale of new physics that lies well below the weak scale would be
required. Potential scenarios are therefore highly constrained, and do not
appear feasible. A direct test is proposed to set bounds on such effects.
