  We present observations of the 85.69 GHz continuum emission and H42alpha line
emission from the central 30 arcsec within NGC 4945. Both sources of emission
originate from nearly identical structures that can be modelled as exponential
discs with a scale length of ~2.1 arcsec (or ~40 pc). An analysis of the
spectral energy distribution based on combining these data with archival data
imply that 84% +/- 10% of the 85.69 GHz continuum emission originates from
free-free emission. The electron temperature is 5400 +/- 600 K, which is
comparable to what has been measured near the centre of the Milky Way Galaxy.
The star formation rate (SFR) based on the H42alpha and 85.69 GHz free-free
emission (and using a distance of 3.8 Mpc) is 4.35 +/- 0.25 M/yr. This is
consistent with the SFR from the total infrared flux and with previous
measurements based on recombination line emission, and it is within a factor of
~2 of SFRs derived from radio data. The Spitzer Space Telescope 24 micron data
and Wide-field Infrared Survey Explorer 22 micron data yield SFRs ~10x lower
than the ALMA measurements, most likely because the mid-infrared data are
strongly affected by dust attenuation equivalent to A_V=150. These results
indicate that SFRs based on mid-infrared emission may be highly inaccurate for
dusty, compact circumnuclear starbursts.
