  The formation of positive ions of antihydrogen $\bar{\rm{H}}^+$ via the three
body reaction (i) $\rm{e}^+ + \rm{e}^- + \bar{\rm{H}} \rightarrow \rm{e}^- +
\bar{\rm{H}}^+$ is considered. In reaction (i), free positrons $\rm{e}^+$ are
incident on antihydrogen $\bar{\rm{H}}$ embedded in a gas of low-energy ($\sim
$ meV) electrons and, due to the positron-electron interaction, a positron is
attached to $\bar{\rm{H}}$ whereas an electron carries away the energy excess.
We compare reaction (i) with two radiative attachment mechanisms. One of them
is (ii) spontaneous radiative attachment, in which the ion is formed due to
spontaneous emission of a photon by a positron incident on $\bar{\rm{H}}$. The
other is (iii) two-center dileptonic attachment which takes place in the
presence of a neighboring atom B and in which an incident positron is attached
to $\bar{\rm{H}}$ via resonant transfer of energy to B with its subsequent
relaxation through spontaneous radiative decay. It is shown that reaction (i)
can strongly dominate over mechanisms (ii) and (iii) for positron energies
below $0.1$ eV. It is also shown that at the energies considered reaction (i)
will not be influenced by annihilation and that the reaction $\rm{e}^+ +
\rm{e}^+ + \bar{\rm{H}} \rightarrow \rm{e}^+ + \bar{\rm{H}}^+$ has a
vanishingly small rate compared to reaction (i).
