  A detailed study of the ground state of the Coulomb system $(\al \al e e)$
which corresponds to the $He_2^{2+}$ molecular ion in a magnetic field
$B=0-4.414 \times 10^{13} $ G in parallel configuration (infinitely massive
$\al-$particles are situated along a magnetic field line) is presented. The
variational method is employed using a trial function which includes electronic
correlation in the form $\exp{(\gamma r_{12})}$ where $\gamma$ is a variational
parameter. It is shown that the quantum numbers of the lowest total energy
state depend on the magnetic field strength. It evolves from the spin-singlet
${}^1\Si_g$ metastable state at $0 \leq B \lesssim 0.85$ a.u. to a repulsive
spin-triplet ${}^3\Si_u$ state for $0.85 {a.u.} \lesssim B \lesssim 1100$ a.u.
and, finally, to a strongly bound spin-triplet ${}^3\Pi_u$ state at stronger
fields $B \gtrsim 1100$ a.u. The lowest vibrational energy for the latter case
is calculated.
