We present James Webb Space Telescope (JWST) and Hubble Space Telescope (HST) observations of the afterglow of GRB 221009A, the brightest gamma-ray burst (GRB) ever observed. This includes the first mid-IR spectra of any GRB, obtained with JWST/Near Infrared Spectrograph (0.6-5.5 micron) and Mid-Infrared Instrument (5-12 micron), 12 days after the burst. Assuming that the intrinsic spectral slope is a single power law, with F ( nu ) proportional to nu (-beta ), we obtain beta approximate to 0.35, modified by substantial dust extinction with A ( V ) = 4.9. This suggests extinction above the notional Galactic value, possibly due to patchy extinction within the Milky Way or dust in the GRB host galaxy. It further implies that the X-ray and optical/IR regimes are not on the same segment of the synchrotron spectrum of the afterglow. If the cooling break lies between the X-ray and optical/IR, then the temporal decay rates would only match a post-jet-break model, with electron index p < 2, and with the jet expanding into a uniform ISM medium. The shape of the JWST spectrum is near-identical in the optical/near-IR to X-SHOOTER spectroscopy obtained at 0.5 days and to later time observations with HST. The lack of spectral evolution suggests that any accompanying supernova (SN) is either substantially fainter or bluer than SN 1998bw, the proto-type GRB-SN. Our HST observations also reveal a disk-like host galaxy, viewed close to edge-on, that further complicates the isolation of any SN component. The host galaxy appears rather typical among long-GRB hosts and suggests that the extreme properties of GRB 221009A are not directly tied to its galaxy-scale environment.