  Context. Hydrogen ionized by young, high-mass stars in starburst galaxies
radiates radio recombination lines (RRLs), whose strength can be used as a
diagnostic of the ionization rate, conditions and gas dynamics in the
starburst, without problems of dust obscuration. However, the lines are weak
and only few extragalactic starburst systems have been detected.
  Aims. We aimed to increase the number of known starburst systems with
detectable RRLs for detailed studies.
  Methods. We searched for the RRLs H91alpha and H92alpha at 8 GHz in the
nearby Seyfert galaxy NGC 4945 using the ATCA with resolution of 3". This
yielded a detection from which we derived conditions in the starburst.
  Results. We detected RRLs from the nucleus of NGC 4945 with a peak strength
of 17.8 mJy, making it the strongest extragalactic RRL emitter known at this
frequency. The line emission from NGC 4945 can be matched by a model consisting
of a collection of 10 to 300 H II regions with temperatures of 5000 K,
densities of 10^3 cm^-3 to 10^4 cm^-3 and a total effective diameter of 2 pc to
100 pc. The Lyman continuum production rate required to maintain the ionization
requires 2000 to 10000 O5 stars to be produced in the starburst, inferring a
star formation rate of 2 Msun/yr to 8 Msun/yr. We resolved the rotation curve
within the central 70 pc region and this is well described by a set of rotating
rings that were coplanar and edge on with a simple flat rotation curve. The
rotation speed of 120 km/s within the central 1" (19 pc) radius infers an
enclosed mass of 3 x 10^7 Msun.
  Conclusion. We discovered RRLs from NGC 4945. It is the strongest known
extragalactic RRL emitter and is suited to high-quality spectroscopic study. We
resolved the dynamics of the ionized gas in the central 70 pc and derived
conditions and star formation rates in the ionized gas.
