  Using a data sample collected with the BESIII detector operating at the
BEPCII storage ring, we observe a new neutral state $Z_c(3900)^{0}$ with a
significance of $10.4\sigma$. The mass and width are measured to be
$3894.8\pm2.3\pm3.2$ MeV/$c^2$ and $29.6\pm8.2\pm8.2$~MeV, respectively, where
the first error is statistical and the second systematic. The Born cross
section for $e^+e^-\to\pi^0\pi^0 J/\psi$ and the fraction of it attributable to
$\pi^0 Z_c(3900)^{0}\to\pi^0\pi^0 J/\psi$ in the range $E_{cm}=4.19-4.42$ GeV
are also determined. We interpret this state as the neutral partner of the
four-quark candidate $Z_c(3900)^\pm$.
