  We study the electron-positron system in a strong magnetic field using the
differential Bethe-Salpeter equation in the ladder approximation. We derive the
fully relativistic two-dimensional form that the four-dimensional
Bethe-Salpeter equation takes in the limit of asymptotically strong constant
and homogeneous magnetic field. An ultimate value for the magnetic field is
determined, which provides the full compensation of the positronium rest mass
by the binding energy in the maximum symmetry state and vanishing of the energy
gap separating the electron-positron system from the vacuum. The compensation
becomes possible owing to the falling to the center phenomenon that occurs in a
strong magnetic field because of the dimensional reduction. The solution of the
Bethe-Salpeter equation corresponding to the vanishing energy-momentum of the
electron-positron system is obtained.
