  We present the results of our observations of the early stages of the
2012--2013 outburst of the transient black hole X-ray binary (BHXRB), Swift
J1745$-$26, with the VLA, SMA, and JCMT (SCUBA--2). Our data mark the first
multiple-band mm & sub-mm observations of a BHXRB. During our observations the
system was in the hard accretion state producing a steady, compact jet. The
unique combination of radio and mm/sub-mm data allows us to directly measure
the spectral indices in and between the radio and mm/sub-mm regimes, including
the first mm/sub-mm spectral index measured for a BHXRB. Spectral fitting
revealed that both the mm (230 GHz) and sub-mm (350 GHz) measurements are
consistent with extrapolations of an inverted power-law from contemporaneous
radio data (1--30 GHz). This indicates that, as standard jet models predict, a
power-law extending up to mm/sub-mm frequencies can adequately describe the
spectrum, and suggests that the mechanism driving spectral inversion could be
responsible for the high mm/sub-mm fluxes (compared to radio fluxes) observed
in outbursting BHXRBs. While this power-law is also consistent with
contemporaneous optical data, the optical data could arise from either jet
emission with a jet spectral break frequency of $\nu_{{\rm
break}}\gtrsim1\times10^{14}\,{\rm Hz}$ or the combination of jet emission with
a lower jet spectral break frequency of $\nu_{{\rm
break}}\gtrsim2\times10^{11}\,{\rm Hz}$ and accretion disc emission. Our
analysis solidifies the importance of the mm/sub-mm regime in bridging the
crucial gap between radio and IR frequencies in the jet spectrum, and justifies
the need to explore this regime further.
