  In 1954, Vogt and Wannier (Phys. Rev. 95, 1190) predicted that the capture
rate of a polarizable neutral atom or molecule by an ion should increase by a
factor of two compared to the classical Langevin rate as the collision energy
approaches zero. This prediction has not been verified experimentally. The
H$_2^+$ + H$_2$ reaction is ideally suited to observe this effect, because the
small reduced mass makes quantum effects related to s-wave scattering
observable at higher collision energies than in other systems. Moreover, the
reaction rate for this barrierless, strongly exothermic reaction follows the
classical Langevin capture model down to cold-collision conditions (about
$k_\mathrm{B} \cdot~1\,\mathrm{K}$) and is not affected by short-range
interactions. Below this temperature, a strong enhancement of the reaction rate
resulting from charge--quadrupole interaction between H$_2^+$ and ground-state
ortho H$_2$ ($J=1$) was observed. Here we present an experimental study of the
reaction of H$_2^+$ and para H$_2$ ($J=0$), which has no dipole and no
quadrupole moments, at collision energies below $k_\mathrm{B}\cdot
1\,\mathrm{K}$. We observe an enhancement at the lowest collision energies
which is attributed to the quantum enhancement predicted by Vogt and Wannier.
Measurements of the reaction of HD$^+$ with HD support this conclusion.
