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 10.4σ. The mass and width are measured to be 3894.8±2.3±3.2 MeV/c^{2} and 29.6±8.2±8.2 MeV, respectively, where the first error is statistical and the second systematic. The Born cross section for e^{+}e^{-}→π^{0}π^{0}J/ψ and the fraction of it attributable to π^{0}Z_{c}(3900)^{0}→π^{0}π^{0}J/ψ in the range E_{c.m.}=4.19-4.42 GeV are also determined. We interpret this state as the neutral partner of the four-quark candidate Z_{c}(3900)^{±}.