  We measure the branching fraction and time-dependent $CP$-violating asymmetry
for $B^0\to J/\psi\pi^0$ decays using a data sample of 711 ${\rm fb^{-1}}$
collected on the $\Upsilon(4S)$ resonance by the Belle experiment running at
the KEKB $e^+e^-$ collider. The branching fraction is measured to be
$\mathcal{B}(B^0\to J/\psi\pi^0) = [1.62\,\pm 0.11~({\rm stat}) \pm0.07~({\rm
syst})]\times 10^{-5}$, which is the most precise measurement to date. The
measured $CP$ asymmetry parameters are $\mathcal{S} = -0.59\,\pm 0.19~({\rm
stat}) \pm 0.03~({\rm syst})$ and $\mathcal{A} = -0.15\,\pm 0.14~({\rm
stat})\,^{+0.04}_{-0.03}~({\rm syst})$. The mixing-induced $CP$ asymmetry
($\mathcal{S}$) differs from the case of no $CP$ violation by 3.0 standard
deviations, and the direct $CP$ asymmetry ($\mathcal{A}$) is consistent with
zero.
