We present a theoretical study of dissociative and nondissociative photoionization of H-2 from the E,F-1 Sigma(+)(g)(upsilon,J=0) excited state for upsilon=0-9 in the photon energy range 3 to 14 eV. We have found that, for most initial upsilon's, dissociative ionization is the dominant process for photon energies well above the dissociative ionization threshold. In this photon energy range, resonance structures arising from autoionization of the Q(1) doubly excited states of H-2 are observed. Cross sections differential in the energy of the remaining H-2(+) ion or in the proton kinetic energy are analyzed in detail for a photon energy of 6.4 eV corresponding to the wavelength of an ArF excimer laser. Comparison of our results with the available experimental measurements is good. We show, however, that contribution of the Q(1) doubly excited states to the measured dissociative ionization cross section is more important than originally believed.