We present a study of the molecular photoionization and photodissociation processes in molecular hydrogen occurring after one-photon absorption from various rovibrational levels (v＇=3-22, J＇ =0-3) of the B (1)Sigma(u)(+) (1s sigma(g))(2p sigma(u)) state using resonance-enhanced multiphoton ionization in combination with high-resolution photoelectron spectroscopy (REMPI-PES). For one-photon absorption from the v＇=3-8 levels, molecular photoionization competes with photodissociation into a ground-state atom and an atom in an n=2 excited state. A detailed comparison of the photoelectron spectra obtained via different rotational branches and vibrational levels strongly indicates that singly excited bound (1)Sigma(g)(+) and (1)Pi(g), Rydberg states at the four-photon level exert a significant influence on the final state distributions of H-2(+). In contrast, one-photon absorption from the v＇=9 and higher levels leads almost exclusively to dissociation into a ground-state atom and an excited-state atom with n> 2. Excited atomic fragments are ionized in a one-photon absorption step, and excited-atom distributions over the energetically allowed values of the principal quantum number I? are obtained. Simulations of these distributions suggest that excitation of dissociative continua of bound (1)Sigma(g)(+) (1s sigma(g))(nd sigma(g)), 1 Sigma(g)(+) (1s sigma(g))(nd sigma(g)), and (1)Pi(g)(1s sigma(g))(nd pi(g)) Rydberg E a g states may dominate over excitation of dissociative doubly excited (1)Sigma(g)(+)(2p sigma(u))(np sigma(u)) and (1)Pi(g)(2p sigma(u))(np pi(u)) states when considering the dissociation dynamics after one-photon absorption from the v＇greater than or equal to 9 levels of the B-state.