  We describe the astrometric reduction of images obtained with the FORS2/VLT
camera in the framework of an astrometric planet search around 20
M/L-transition dwarfs. We present the correction of systematic errors, the
achieved astrometric performance, and a new astrometric catalogue containing
the faint reference stars in 20 fields located close to the Galactic plane. We
detected three types of systematic errors in the FORS2 astrometry: the relative
motion of the camera's two CCD chips, errors that are correlated in space, and
an error contribution of yet unexplained origin. The relative CCD motion has
probably a thermal origin and usually is 0.001-0.010 px (~0.1-1 mas), but
sometimes amounts to 0.02-0.05 px (3-6 mas). This instability and
space-correlated errors are detected and mitigated using reference stars. The
third component of unknown origin has an amplitude of 0.03-0.14 mas and is
independent of the observing conditions. We find that a consecutive sequence of
32 images of a well-exposed star over 40 min at 0.6" seeing results in a median
r.m.s. of the epoch residuals of 0.126 mas. Overall, the epoch residuals are
distributed according to a normal law with a chi2~1. We compiled a catalogue of
12000 stars with I-band magnitudes of 16-22 located in 20 fields, each covering
~2x2'. It contains I-band magnitudes, ICRF positions with 40-70 mas precision,
and relative proper motions and absolute trigonometric parallaxes with a
precision of 0.1 mas/yr and 0.1 mas at the bright end, respectively.
