Many X-ray binaries remain undetected in the mid-infrared, a regime where emission from their compact jets is likely to dominate. Here, we report the detection of the black hole binary GX 339-4 with the Wide-field Infrared Survey Explorer (WISE) during a very bright, hard accretion state in 2010. Combined with a rich contemporaneous multiwavelength data set, clear spectral curvature is found in the infrared, associated with the peak flux density expected from the compact jet. An optically thin slope of ~ -0.7 and a jet radiative power of >6 × 1035 erg s-1 (d/8 kpc)2 are measured. A ~24 hr WISE light curve shows dramatic variations in mid-infrared spectral slope on timescales at least as short as the satellite orbital period ~95 minutes. There is also significant change during one pair of observations spaced by only 11 s. These variations imply that the spectral break associated with the transition from self-absorbed to optically thin jet synchrotron radiation must be varying across the full wavelength range of ~3-22 μm that WISE is sensitive to, and more. Based on four-band simultaneous mid-infrared detections, the break is constrained to frequencies of ≈4.6+3.5 - 2.0 × 1013 Hz in at least two epochs of observation, consistent with a magnetic field B ≈ 1.5(± 0.8) × 104 G assuming a single-zone synchrotron emission region. The observed variability implies that either B or the size of the acceleration zone above the jet base is being modulated by factors of ~10 on relatively short timescales.