  We propose and discuss a new experimental approach to measure the centroid
shift induced by gravitational microlensing in the images of lensed quasars
(astrometric microlensing). Our strategy is based on taking the photocenter of
a region in the quasar large enough as to be insensitive to microlensing as
reference to measure the centroid displacement of the continuum. In this way,
single-epoch measurements of astrometric microlensing can be performed. Using
numerical simulations, we show that, indeed, the centroid shift monotonically
decreases as the size of the emitting region increases, and only for relatively
large regions, like the broad line region (BLR), does the centroid shift become
negligible. This opens interesting possibilities to study the stratification of
the different emitters in the accretion disk and the BLR. We estimate the
amplitude of the centroid shifts for 79 gravitationally lensed images and study
more thoroughly the special cases Q2237+030 A, RXJ1131-1231 A, PG1115+080 A2
and SDSS J1004+4112 A. We propose to use spectro-astrometry to simultaneously
obtain the photocenters of the continuum and of different emission line regions
since, with the precision of forthcoming instruments, astrometric microlensing
by $\sim 1 M_\odot$ mass microlenses may be detected in many quasar lensed
images. When we consider more massive micro/millilenses, $M\gtrsim 10 M_\odot$,
often proposed as the constituents of dark matter, the BLR becomes sensitive to
microlensing and can no longer be used as a positional reference to measure
centroid shifts. Differential microlensing between the images of a lensed
quasar along several epochs should be used instead.
