We present results from a fifteen-month campaign of high-cadence (~ 3 days) mid-infrared Spitzer and optical (B and V ) monitoring of the Seyfert 1 galaxy NGC 6418, with the objective of determining the characteristic size of the dusty torus in this active galactic nucleus (AGN). We find that the 3.6 $mu$m and 4.5 $mu$m flux variations lag behind those of the optical continuum by $37.2^{+2.4}_{-2.2}$ days and $47.1^{+3.1}_{-3.1}$ days, respectively. We report a cross-correlation time lag between the 4.5 $mu$m and 3.6 $mu$m flux of $13.9^{+0.5}_{-0.1}$ days. The lags indicate that the dust emitting at 3.6 $mu$m and 4.5 $mu$m is located at a distance of approximately 1 light-month (~ 0.03 pc) from the source of the AGN UV-optical continuum. The reverberation radii are consistent with the inferred lower limit to the sublimation radius for pure graphite grains at 1800 K, but smaller by a factor of ~ 2 than the corresponding lower limit for silicate grains; this is similar to what has been found for near-infrared (K-band) lags in other AGN. The 3.6 and 4.5 $mu$m reverberation radii fall above the K-band $tau propto L^{0.5}$ size-luminosity relationship by factors $lesssim 2.7$ and $lesssim 3.4$, respectively, while the 4.5 $mu$m reverberation radius is only 27% larger than the 3.6 $mu$m radius. This is broadly consistent with clumpy torus models, in which individual optically thick clouds emit strongly over a broad wavelength range.
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