The nucleus of the nearby spiral galaxy M81 was observed at 8.3 GHz with a global VLBI array at 20 epochs over four and a half years, with a linear resolution at the source of about 2000 AU or 0.01 pc. Phase-referenced mapping with respect to the geometric center of supernova 1993J in the same galaxy enabled us to find, with a standard error of about 600 AU, a stationary point in the source south-east of the brightness peak. We identify this point as the location of the core and the putative black hole at the gravitational center of the galaxy. The 2σ upper bound on the core's average velocity on the sky is ≤ 40 µas yr −1 or ≤ 730 km s −1 (relative to the center of SN1993J, in excess of galactic rotation). A short, one-sided jet extends towards the north-east from the core, in projection approximately in line with the rotation axis of the galaxy, and towards the well-known extended emission 1 kpc further out. The orientation of the jet varies smoothly, with timescales of about one year, and an rms of 6 • , around its mean position angle of 50 • . Occasionally the jet appears to bend to the east. The length of the jet is only about 1 mas (3,600 AU), and varies with an rms of about 20% from epoch to epoch. The inferred speeds are below 0.08c. The total flux density of the core-jet varies erratically, changing on occasion by a factor of two over a few weeks, without any significant changes in the source size and orientation. The inferred velocity of the plasma flow is > 0.25c. The results are consistent with a model in which plasma condensations with short lifetimes are ejected relativistically from the core on a timescale of less than a few weeks. The condensations travel along a tube whose pattern and geometry are also variable but only on a timescale of about one year. The central engine of M81 has qualitative similarities to those of powerful active galactic nuclei of radio galaxies and quasars, and may also represent in power and size a scaled-up version of the largely hidden nucleus in our own Galaxy.