Using data obtained by the DIRBE instrument on the COBE spacecraft, we present the mean 3.5-240 μm spectrum of high-latitude dust. Combined with a spectrum obtained by the FIRAS instrument, these data represent the most comprehensive wavelength coverage of dust in the diffuse interstellar medium, spanning the 3.5-1000 μm wavelength regime. At wavelengths shorter than ~60 μm the spectrum shows an excess of emission over that expected from dust heated by the local interstellar radiation field and radiating at an equilibrium temperature. The DIRBE data thus extend the observations of this excess, first detected by the IRAS satellite at 25 and 12 μm, to shorter wavelengths. The excess emission arises from very small dust particles undergoing temperature fluctuations. However, the 3.5-4.9 μm intensity ratio cannot be reproduced by very small silicate or graphite grains. The DIRBE data strongly suggest that the 3.5-12 μm emission is produced by carriers of the ubiquitous 3.3, 6.2, 7.7, 8.6, and 11.3 μm solid state emission features that have been detected in a wide variety of astrophysical objects. The carriers of these features have been widely identified with polycyclic aromatic hydrocarbons (PAHs). Our dust model consists of a mixture of PAH molecules and bare astronomical silicate and graphite grains with optical properties given by Draine & Lee. We obtain a very good fit to the DIRBE spectrum, deriving the size distribution, abundances relative to the total hydrogen column density, and relative contribution of each dust component to the observed IR emission. At wavelengths above 140 μm the model is dominated by emission from T ≈ 17-20 K graphite and 15-18 K silicate grains. The model provides a good fit to the FIRAS spectrum in the 140-500 μm wavelength regime but leaves an excess Galactic emission component at 500-1000 μm. The nature of this component is still unresolved. We find that (C/H) is equal to (7.3 ± 2.2) × 10-5 for PAHs and equal to (2.5 ± 0.8) × 10-4 for graphite grains, requiring about 20% of the cosmic abundance of carbon to be locked up in PAHs, and about 70% in graphite grains [we adopt (C/H)☉ = 3.6 × 10-4]. The model also requires all of the available magnesium, silicon, and iron to be locked up in silicates. The power emitted by PAHs is 1.6 × 10-31 W per H atom, by graphite grains 3.0 × 10-31 W per H atom, and by silicates 1.4 × 10-31 W per H atom, adding up to a total infrared intensity of 6.0 × 10-31 W per H atom, or ~2 L☉ M. The [C II] 158 μm line emission detected by the FIRAS provides important information on the gas phase abundance of carbon in the diffuse ISM. The 158 μm line arises predominantly from the cold neutral medium (CNM) and shows that for typical CNM densities and temperatures C+/H = (0.5-1.0) × 10-4, which is ~14%-28% of the cosmic carbon abundance. The remaining carbon abundance in the CNM, which must be locked up in dust, is about equal to that required to provide the observed IR emission, consistent with notion that most (75%) of this emission arises from the neutral component of the diffuse ISM. The model provides a good fit to the general interstellar extinction curve. However, at UV wavelengths it predicts a larger extinction. The excess extinction may be the result of the UV properties adopted for the PAHs. If real, the excess UV extinction may be accounted for by changes in the relative abundances of PAHs and carriers of the 2200 A extinction bump.