  An accurate study of the lowest $1\sigma_g$ and the low-lying excited
$1\sigma_u$, $1\pi_{u,g}$, $1\delta_{g,u}$ electronic states of the exotic
molecular ion $H_3^{2+}$ in linear configuration parallel to a magnetic field
is carried out. The magnetic field ranges from $10^{10}$ G up to $4.414 \times
10^{13}$ G where non-relativistic considerations are justified. The variational
method is exploited and the {\it same} trial function is used for different
magnetic fields. It is shown that the states of positive $z$-parity $1\sigma_g,
1\pi_u, 1\delta_{g}$ are bound states of the $H_3^{2+}$ exotic ion for all
magnetic fields studied. We also demonstrate that for magnetic fields $B\gtrsim
2.35\times 10^{12}$ G the potential energy surface well corresponding to the
$1\sigma_g$ state contains at least one longitudinal vibrational state. It is
also shown that the negative $z$-parity states $1\sigma_u, 1\pi_g,
1\delta_{u}$, are purely repulsive in the whole range of magnetic fields
studied, $B=10^{10}- 4.414 \times 10^{13}$ G.
