Using a sample of (10.09±0.04)×10^{9} J/ψ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay J/ψ→γηη^{'} is performed. The first observation of an isoscalar state with exotic quantum numbers J^{PC}=1^{-+}, denoted as η_{1}(1855), is reported in the process J/ψ→γη_{1}(1855) with η_{1}(1855)→ηη^{'}. Its mass and width are measured to be (1855±9_{-1}^{+6}) MeV/c^{2} and (188±18_{-8}^{+3}) MeV, respectively, where the first uncertainties are statistical and the second are systematic, and its statistical significance is estimated to be larger than 19σ.