We report on the first cross section measurements for charged current coherent pion production by neutrinos and antineutrinos on argon. These measurements are performed using the ArgoNeuT detector exposed to the NuMI beam at Fermilab. The cross sections are measured to be $2.6^{+1.2}_{-1.0}(stat)^{+0.3}_{-0.4}(syst) times 10^{-38} textrm{cm}^{2}/textrm{Ar}$ for neutrinos at a mean energy of $9.6$ GeV and $5.5^{+2.6}_{-2.1}(stat)^{+0.6}_{-0.7}(syst) times 10^{-39} textrm{cm}^{2}/textrm{Ar}$ for antineutrinos at a mean energy of $3.6$ GeV.