  Molecular vibrational transitions are prime candidates for model-independent
searches for variation of the proton-to-electron mass ratio. Searches for
present-day variation achieve highest sensitivity with deep molecular
potentials. We identify several high-sensitivity transitions in the deeply
bound ${\rm O}_2^+$ molecular ion. These transitions are electric-dipole
forbidden and thus have narrow linewidths. The most sensitive transitions take
advantage of an accidental degeneracy between vibrational states in different
electronic potentials. We suggest experimentally feasible routes to a
measurement with uncertainty exceeding current limits on present-day variation
in $m_p/m_e$.
