The decay channel J/psi --> gamma(pi)(+)pi(-)eta is analyzed using a sample of 5.8 x 10(7) J/psi events collected with the BESII detector. A resonance, the Chi(1835), is observed in the pi(+)pi(-)eta invariant-mass spectrum with a statistical significance of 7.7 sigma. A fit with a Breit-Wigner function yields a mass M = 1833.7 +/- 6.1(stat) +/- 2.7(syst) MeV/c(2), a width Tau = 67.7 +/- 20.3(stat) +/- 7.7(syst) MeV/c(2), and a product branching fraction B(J/psi --> gammaChi) . B(Chi --> pi(+)pi(-)eta) = [2.2 +/- 0.4(stat) +/- 0.4(syst)] x 10(-4). The mass and width of the Chi(1835) are not compatible with any known meson resonance. Its properties are consistent with expectations for the state that produces the strong pp mass threshold enhancement observed in the J/psi --> gammapp process at BESII.