  The $B^0_s\to J/\psi f_0(980)$ decay offers an interesting experimental
alternative to the well-known $B^0_s\to J/\psi \phi$ channel for the search of
CP-violating New-Physics contributions to $B^0_s$--$\bar B^0_s$ mixing. As the
hadronic structure of the $f_0(980)$ has not yet been settled, we take a
critical look at the implications for the relevant observables and address
recent experimental data. It turns out that the effective lifetime of $B^0_s\to
J/\psi f_0(980)$ and its mixing-induced CP asymmetry $S$ are quite robust with
respect to hadronic effects and thereby allow us to search for a large
CP-violating $B^0_s$--$\bar B^0_s$ mixing phase $\phi_s$, which is tiny in the
Standard Model. However, should small CP violation, i.e. in the range
$-0.1\lsim S\lsim 0$, be found in $B^0_s\to J/\psi f_0(980)$, it will be
crucial to constrain hadronic corrections in order to distinguish possible
New-Physics effects from the Standard Model. We point out that $B^0_d\to J/\psi
f_0(980)$, which has not yet been measured, is a key channel in this respect
and discuss the physics potential of this decay.
