We present the discovery of fast infrared/X-ray correlated variability in the black hole transient GX 339-4. The source was observed with subsecond time resolution simultaneously with Very Large Telescope/Infrared Spectrometer And Array Camera and Rossi X-ray Timing Explorer/Proportional Counter Array in 2008 August, during its persistent low-flux highly variable hard state. The data show a strong correlated variability, with the infrared emission lagging the X-ray emission by 100ms. The short time delay and the nearly symmetric cross-correlation function, together with the measured brightness temperature of ~2.5 × 106K, indicate that the bright and highly variable infrared emission most likely comes from a jet near the black hole. Under standard assumptions about jet physics, the measured time delay can provide us a lower limit of Γ > 2 for the Lorentz factor of the jet. This suggests that jets from stellar-mass black holes are at least mildly relativistic near their launching region. We discuss implications for future applications of this technique. Based on observations made with European Southern Observatory (ESO) Telescopes at the Paranal Observatory under programme ID 281.D-5034. E-mail: p.casella [at] soton.ac.uk (p[dot]casella[at]soton[dot]ac[dot]uk)