  We study the question about existence i.e. stability with respect to
dissociation of the spin-quartet, permutation- and reflection-symmetric
${}^4(-3)^+_g$ ($S_z=-3/2, M=-3$) state of the $(\alpha\alpha e e e)$ Coulomb
system: the ${\rm He}_2^+$ molecular ion, placed in a magnetic field $0 \le B
\le 10000$ a.u. We assume that the $\alpha$-particles are infinitely massive
(Born-Oppenheimer approximation of zero order) and adopt the parallel
configuration, when the molecular axis and the magnetic field direction
coincide, as the optimal configuration. The study of the stability is performed
variationally with a physically adequate trial function. To achieve this goal,
we explore several Helium-contained compounds in strong magnetic fields, in
particular, we study the spin-quartet ground state of ${\rm He}^-$ ion, and the
ground (spin-triplet) state of the Helium atom, both for a magnetic field in
$100 \leq B\leq 10000$ a.u. The main result is that the ${\rm He}_2^+$
molecular ion in the state ${}^4(-3)^+_g$ is stable towards all possible decay
modes for magnetic fields $B \gtrsim 120$ a.u. and with the magnetic field
increase the ion becomes more tightly bound and compact with a cigar-type form
of electronic cloud. At $B=1000$ a.u., the dissociation energy of ${\rm
He}_2^+$ into ${\rm He}^- + \alpha$ is $\sim 701.8$ eV and the dissociation
energy for the decay channel to ${\rm He} + \alpha + e $ is $\sim 729.1$ eV,
latter both energies are in the energy window for one of the observed
absorption features of the isolated neutron star 1E1207.4-5209.
