  We have investigated the chemistry of ${\rm C + H_3^+}$ forming CH$^+$,
CH$_2^+$, and CH$_3^+$. These reactions are believed to be some of the key
gas-phase astrochemical processes initiating the formation of organic molecules
in molecular clouds. For this work we have constructed a novel merged
fast-beams apparatus which overlaps a beam of molecular ions onto a beam of
ground-term neutral atoms. Here we present cross section data for forming
CH$^+$ and CH$_2^+$ at collision energies from $\approx 9$ meV to $\approx20$
and 3 eV, respectively. Using these data we have derived thermal rate
coefficients for reaction temperatures from $\approx72$ K to $\approx2.3 \times
10^5$ and $3.4 \times 10^4$ K, respectively. For the formation of CH$_3^+$ we
are able only to put an upper limit on the rate coefficient. Our results for
CH$^+$ and CH$_2^+$ are in good agreement with the mass-scaled results from a
previous ion trap study of ${\rm C + D_3^+}$ at a reaction temperature of $\sim
1000$ K. At molecular cloud temperatures our thermal rate coefficient for
forming CH$^+$ lies a factor of $\sim 2-4$ below the Langevin rate coefficient
currently given in astrochemical databases and below the published
semi-classical calculations. Our results for CH$_2^+$ formation are a factor of
$\sim 26$ above the semi-classical results. Astrochemical databases do not
currently include this channel.
