Using a data sample collected with the BESIII detector operating at the BEPCII storage ring, we observe a new neutral state Zc(39000)0 with a significance of 10.4σ. The mass and width are measured to be 3894.8 ± 2.3 ± 3.2 MeV=c2 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− → π0π0J=ψ and the fraction of it attributable to ψ0Zc(39000)0 → π0π0J=ψ in the range Ec.m. 4.19−4.42 GeV are also determined. We interpret this state as the neutral partner of the four-quark candidate Zc(3900)±.