We have studied the dissociative recombination (DR) of molecular hydrogen ions with slow electrons over a range of collision energies from 0 to 400 meV. By employing a pulsed expansion source for rotational cooling and by exploiting super elastic collisions with near-0-eV electrons in a heavy ion storage ring for vibrational cooling, we observe a highly structured DR cross section, comparable to that reported for HD+. Using para-hydrogen-enriched ion beams, we identify for the first time features in the DR cross sections attributed to nu=0, J=even molecules (para-H2) and nu=0, J=odd (ortho-H2) molecules, separately. Indications are given that para levels have different DR rate coefficients from ortho levels for the first four vibrational levels at near-0-eV collisions.