We have constructed a three-dimensional model of the Galactic large-scale infrared emission from dust associated with the molecular (H2), neutral atomic (H I), and extended low-density (ne ~ 1-100 cm-3) ionized (H II) gas phases of the interstellar medium. The model incorporates a three-dimensional map of the molecular and neutral atomic hydrogen gas distributions, derived from available 12CO and H I surveys by using the radial velocity information in the spectral lines as a distance indicator, and available 5 and 19 GHz radio continuum surveys to trace the column density of ionized gas. We use the model to decompose the COBE5 Diffuse Infrared Background Experiment (DIRBE) 12-240 μm observations of the Galactic plane region (|b| <= 5°), from which the zodiacal light and stellar emission have been subtracted, into distinct emission components associated with each gas phase within selected ranges of Galactocentric distance. An interstellar dust model is fitted to the resulting infrared spectra to derive the following quantities within each Galactocentric distance interval: (1) the abundance and equilibrium temperature of the large dust grain component within each gas phase; (2) estimates of the abundance of very small (<200 Å) transiently heated dust grains and polycyclic aromatic hydrocarbon (PAH) molecules; and (3) constraints on various model parameters, such as the energy density of the ambient interstellar radiation field, which heats the dust within the H I gas phase.Our results show steep negative Galactocentric gradients in the equilibrium temperature of the large dust grain component within the H I, H2, and H II gas phases, the Galaxy's ambient interstellar radiation field, and the dust-to-gas mass ratio for each gas phase. The intensity of the ambient interstellar radiation field increases by a factor of ~3 between the solar circle (8.5 kpc) and the molecular ring at a Galactocentric distance of ~5 kpc. The dust abundance gradient of (-0.05 +/- 0.03) dex kpc-1 is equivalent, within the uncertainties, to the metallicity gradient in the Galactic disk. The derived emission spectra are consistent with a model in which very small transiently heated dust grains and PAHs are abundant and the dominant contributors to the mid-infrared (5 μm < λ < 40 μm) luminosity from a Galactocentric distance of 2 kpc out to a Galactocentric distance of at least 12 kpc, and indicate that the relative abundance of the PAHs is significantly higher in the outer region of the Galactic disk than inside the solar circle. We combine the results of our decomposition algorithm with the results of a study of optical extinction at high Galactic latitude to derive the radial distribution of optical opacity in the Galactic disk and find that our Galaxy would be effectively transparent [AB(total Galaxy) < 0.2 mag] to an external observer viewing it at a low inclination (i < 30°). All of the Galactic infrared emission observed by the DIRBE can be accounted for by dust associated with gas that is detected by current radio surveys, refuting the recent suggestion that a large fraction of the dynamically inferred hidden mass in spiral galaxies may be due to unseen gas and stars in the disk of the galaxies.