ABSTRACT We report on Atacama Large Millimeter Array (ALMA) continuum observations of the black hole X-ray binary A0620–00 at an X-ray luminosity nine orders of magnitude sub-Eddington. The system was significantly detected at 98 GHz (at 44 ± 7 $μ$Jy) and only marginally at 233 GHz (20 ± 8 $μ$Jy), about 40 d later. These results suggest either an optically thin sub-mm synchrotron spectrum, or highly variable sub-mm jet emission on month time-scales. Although the latter appears more likely, we note that, at the time of the ALMA observations, A0620–00 was in a somewhat less active optical-IR state than during all published multiwavelength campaigns when a flat-spectrum, partially self-absorbed jet has been suggested to extend from the radio to the mid-IR regime. Either interpretation is viable in the context of an internal shock model, where the jet’s spectral shape and variability are set by the power density spectrum of the shells’ Lorentz factor fluctuations. While strictly simultaneous radio–mm-IR observations are necessary to draw definitive conclusions for A0620–00, the data presented here, in combination with recent radio and sub-mm results from higher luminosity systems, demonstrate that jets from black hole X-ray binaries exhibit a high level of variability – either in flux density or intrinsic spectral shape, or both – across a wide spectrum of Eddington ratios. This is not in contrast with expectations from an internal shock model, where lower jet power systems can be expected to exhibit larger fractional variability owing to an overall decrease in synchrotron absorption.