We present new observations of the black hole X-ray binary A0620-00 using the Mid-Infrared (MIR) Instrument on the James Webb Space Telescope, during a state where the X-ray luminosity is 9 orders of magnitude below Eddington, and coordinated with radio, near-infrared, and optical observations. The goal is to understand the nature of the excess MIR emission originally detected by Spitzer redward of 8 μm. The stellar-subtracted MIR spectrum is well modeled by a power law with a spectral index of α = 0.72 ± 0.01, where the flux density scales with frequency as Fν ∝ να. The spectral characteristics, along with rapid variability—a 40% flux flare at 15 μm and 25% achromatic variability in the 5-12 μm range—rule out a circumbinary disk as the source of the MIR excess. The Low Resolution Spectrometer reveals a prominent emission feature at 7.5 μm, resulting from the blend of three hydrogen recombination lines. While the contribution from partially self-absorbed synchrotron radiation cannot be ruled out, we argue that thermal bremsstrahlung from a warm (a few tens of thousands of Kelvin) wind accounts for the MIR excess; the same outflow is responsible for the emission lines. The inferred mass outflow rate indicates that the system’s low luminosity is due to a substantial fraction of the mass supplied by the donor star being expelled through a wind rather than accreted onto the black hole.