  Deuterated molecules are important chemical tracers of prestellar and
protostellar cores. Up to now, the titular reaction has been assumed to
contribute to the generation of these deuterated molecules. We have measured
the merged-beams rate coefficient for this reaction as function of the relative
collision energy in the range of about 10 meV to 10 eV. By varying the internal
temperature of the reacting H$_3^+$ molecules, we found indications for the
existence of a reaction barrier. We have performed detailed theoretical
calculations for the zero-point-corrected energy profile of the reaction and
determined a new value for the barrier height of $\approx$ 68 meV. Furthermore,
we have calculated the tunneling probability through the barrier. Our
experimental and theoretical results show that the reaction is essentially
closed at astrochemically relevant temperatures. We derive a thermal rate
coefficient of $<1\times 10^{-12}$ cm$^3$ s$^{-1}$ for temperatures below 75 K
with tunneling effects included and below 155 K without tunneling.
