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 $<1times 10^{-12}$ cm$^3$ s$^{-1}$ for temperatures below 75 K with tunneling effects included and below 155 K without tunneling.
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