  Using a sample of (10.09$\pm$0.04)$\times$10$^{9}$ $J/\psi$ events collected
with the BESIII detector operating at the BEPCII storage ring, a partial wave
analysis of the decay $J/\psi \rightarrow \gamma\eta\eta'$ is performed. The
first observation of an isoscalar state with exotic quantum numbers
$J^{PC}=1^{-+}$, denoted as $\eta_1(1855)$, is reported in the process $J/\psi
\rightarrow \gamma\eta_1(1855)$ with $\eta_1(1855)\rightarrow\eta\eta'$. Its
mass and width are measured to be (1855$\pm$9$_{-1}^{+6}$)MeV/$c^{2}$ and
(188$\pm$18$_{-8}^{+3}$)MeV, respectively, where the first uncertainties are
statistical and the second are systematic, and its statistical significance is
estimated to be larger than 19$\sigma$.
