ABSTRACT We present observations of the pulsar-wind nebula (PWN) region of SNR 0540-69.3. The observations were made with the Atacama Compact Array (ACA) in Bands 4 and 6. We also add radio observations from the Australia Compact Array at 3 cm. For 1.449–233.50 GHz, we obtain a synchrotron spectrum $F_{ν } ∝ ν ^{-α _{ν }}$, with the spectral index αν = 0.17 ± 0.02. To conclude how this joins the synchrotron spectrum at higher frequencies, we include hitherto unpublished AKARI mid-infrared data, and evaluate published data in the ultraviolet (UV), optical, and infrared (IR). In particular, some broad-band filter data in the optical must be discarded from our analysis due to contamination by spectral line emission. For the UV/IR part of the synchrotron spectrum, we arrive at $α _{ν } = 0.87^{+0.08}_{-0.10}$. There is room for 2.5 × 10−3 M⊙ of dust with a temperature of ∼55 K if there are dual breaks in the synchrotron spectrum, one around ∼9 × 1010 Hz and another at ∼2 × 1013 Hz. The spectral index then changes at ∼9 × 1010 Hz from αν = 0.14 ± 0.07 in the radio to $α _{ν } = 0.35^{-0.07}_{+0.05}$ in the millimetre-to-far-IR range. The ACA Band 6 data marginally resolve the PWN. In particular, the strong emission $\text{$\sim$} 1\hbox{$.\!\!^{\prime \prime }$}5$ south-west of the pulsar, seen at other wavelengths, and resolved in the 3 cm data with its 0.″8 spatial resolution, is also strong in the millimetre range. The ACA data clearly reveal the supernova remnant shell ∼20–35 arcsec west of the pulsar, and for the shell we derive αν = 0.64 ± 0.05 for the range 8.6–145 GHz.