We have used the Spitzer Space Telescope Infrared Spectrograph (IRS) to observe the 5-37 micron thermal emission of comet 73P/Schwassmann-Wachmann 3 (SW3), components B and C. We obtained low spectral resolution (R ~ 100) data over the entire wavelength interval, along with images at 16 and 22 micron. These observations provided an unprecedented opportunity to study nearly pristine material from the surface and what was until recently the interior of an ecliptic comet - cometary surface having experienced only two prior perihelion passages, and including material that was totally fresh. The spectra were modeled using a variety of mineral types including both amorphous and crystalline components. We find that the degree of silicate crystallinity, ~ 35%, is somewhat lower than most other comets with strong emission features, while its abundance of amorphous carbon is higher. Both suggest that SW3 is among the most chemically primitive solar system objects yet studied in detail, and that it formed earlier or farther from the sun than the bulk of the comets studied so far. The similar dust compositions of the two fragments suggests that these are not mineralogically heterogeneous, but rather uniform throughout their volumes. Atomic abundances derived from the spectral models indicates a depletion of O compared to solar photospheric values, despite the inclusion of water ice and gas in the models. Atomic C may be solar or slightly sub-solar, but its abundance is complicated by the potential contribution of spectrally featureless mineral species to the portion of the spectra most sensitive to the derication of the C abundance. We find a relatively high bolometric albedo, ~ 0.13 for the dust, considering the large amount of dark carbonaceous material, but consistent with the presence of abundant small particles and strong emission features.