The single-center adiabatic-nuclei convergent close-coupling method is used to investigate positron collisions with molecular hydrogen (H2) in the ground and first vibrationally excited states. Cross sections are presented over the energy range from 1 to 1000 eV for elastic scattering, vibrational excitation, total ionization, and the grand total cross section. The present adiabatic-nuclei positron-H2 scattering length is calculated as A=-2.70a0 for the ground state and A=-3.16a0 for the first vibrationally excited state. The present elastic differential cross sections are also used to "correct" the low-energy grand total cross-section measurements of the Trento group [A. Zecca, Phys. Rev. A 80, 032702 (2009)PLRAAN1050-294710.1103/PhysRevA.80.032702] for the forward-angle-scattering effect. In general, the comparison with experiment is good. By performing convergence studies, we estimate that our Rm=1.448a0 fixed-nuclei results are converged to within ±5% for the major scattering integrated cross sections.