A broad peak is observed at low K+K- invariant mass in J/psi -> K+K- pi(0) decays found in a sample of 5.8 x 10(7) J/psi events collected with the BESII detector. The statistical significance of the broad resonance is much larger than 5 sigma. A partial wave analysis shows that the J(PC) of this structure is 1(--). Its pole position is determined to be [1576(-55)(+49)(stat)(-91)(+98)(syst)] MeV/c(2)-i/2[818(-23)(+22)(stat)(-133)(+64)(syst)] MeV/c(2). These parameters are not compatible with any known meson resonances.