  The exact precession frequency of a freely-precessing test gyroscope is
derived for a $2+1$ dimensional rotating acoustic black hole analogue
spacetime, without making the somewhat unrealistic assumption that the
gyroscope is static. We show that, as a consequence, the gyroscope crosses the
acoustic ergosphere of the black hole with a finite precession frequency,
provided its angular velocity lies within a particular range determined by the
stipulation that the Killing vector is timelike over the ergoregion.
Specializing to the `Draining Sink' acoustic black hole, the precession
frequency is shown to diverge near the acoustic horizon, instead of the
vicinity of the ergosphere. In the limit of an infinitesimally small rotation
of the acoustic black hole, the gyroscope still precesses with a finite
frequency, thus confirming a behaviour analogous to geodetic precession in a
physical non-rotating spacetime like a Schwarzschild black hole. Possible
experimental approaches to detect acoustic spin precession and measure the
consequent precession frequency, are discussed.
