Photoionisation partial cross sections, continuum phaseshifts and angular distributions of photoelectrons for the hydrogen molecular ion ground state, H₂⁺1 sigma _g(² Sigma _g⁺) up to an energy of 20 au have been calculated using a two-dimensional numerical treatment, and an exact one-electron treatment in spheroidal coordinates. The calculations presented extend the range and precision of previous work. The cross section, asymmetry parameter and continuum phaseshifts for the hydrogen molecule have been also calculated using fully numerical two-dimensional continuum functions at the Hartree-Fock level with exchange. The cross section and asymmetry parameter are compared with experiment in both the length and velocity dipole forms, and the effect of exchange is assessed.