We present results from a 15 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 and 4.5{mu}m flux variations lag behind those of the optical continuum by 37.2_-2.2_^+2.4^-days and 47.1_-3.1_^+3.1^-days, respectively. We report a cross-correlation time lag between the 4.5 and 3.6{mu}m flux of 13.9_-0.1_^+0.5^ days. The lags indicate that the dust emitting at 3.6 and 4.5{mu}m is located at a distance ~1-light-month (~0.03pc) 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 1800K, 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 AGNs. 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 <~2.8 and <~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.