  A completely general parametrization of the time-dependent decay rates of the
modes $B_s\to J/\psi K_S$ and $B_d\to J/\psi K_S$ is given, which are related
to each other through the $U$-spin flavour symmetry of strong interactions.
Owing to the interference of current--current and penguin processes, the
$B_s\to J/\psi K_S$ observables probe the angle $\gamma$ of the unitarity
triangle. Using the $U$-spin symmetry, the overall normalization of the $B_s\to
J/\psi K_S$ rate can be fixed with the help of the CP-averaged $B_d\to J/\psi
K_{\rm S}$ rate, providing a new strategy to determine $\gamma$. This
extraction of $\gamma$ is not affected by any final-state-interaction effects,
and its theoretical accuracy is only limited by $U$-spin-breaking corrections.
As a by-product, this strategy allows us to take into account also the penguin
effects in the determination of $\beta$ from $B_d\to J/\psi K_S$, which are
presumably very small, and to predict the direct CP asymmetry arising in this
mode. An analogous strategy is provided by the time-dependent $B_d\to D^+ D^-$
rate, if its overall normalization is fixed through the CP-averaged $B_s\to
D^+_s D^-_s$ rate.
