  We theoretically investigate the single and double ionization of the He atom
by antiproton impact for projectile energies ranging from $3$~keV up to
$1000$~keV. We obtain accurate total cross sections by directly solving the
fully correlated two-electron time-dependent Schr\"odinger equation and by
performing classical trajectory Monte-Carlo calculations. The obtained
quantum-mechanical results are in excellent agreement with the available
experimental data. Along with the total cross sections, we also present the
first fully \textit{ab initio} doubly differential data for single ionization
at 10 and 100~keV impact energies. In these differential cross sections we
identify the binary-encounter peak along with the anticusp minimum.
Furthermore, we also point out the importance of the post-collisional
electron-projectile interaction at low antiproton energies which significantly
suppresses electron emission in the forward direction.
