  The B^0 -> J/\psi K_{S,L} channels are outstanding probes of CP violation. We
have a detailed look at the associated Standard-Model uncertainties, which are
related to doubly Cabibbo-suppressed penguin contributions, and point out that
these usually neglected effects can actually be taken into account
unambiguously through the CP asymmetries and the branching ratio of the B^0 ->
J/\psi pi^0 decay. Using the most recent B-factory measurements, we find a
negative shift of the extracted value of beta, which softens the tension in the
fits of the unitarity triangle. In addition, this strategy can be used to
constrain a possible new-physics phase in B^0-\bar B^0 mixing. The proposed
strategy is crucial to fully exploit the tremendous accuracies for the search
for this kind of new physics that can be achieved at the LHC and future
super-flavour factories.
