  The Gaia early Data Release 3 has delivered exquisite astrometric data for
1.47 billion sources, which is revolutionizing many fields in astronomy. For a
small fraction of these sources, the astrometric solutions are poor, and the
reported values and uncertainties may not apply. Before any analysis, it is
important to recognize and excise these spurious results - this is commonly
done by means of quality flags in the Gaia catalog. Here, we devise a means of
separating 'good' from 'bad' astrometric solutions that is an order of
magnitude cleaner than any single flag: 99.3% pure and 97.3% complete, as
validated on our test data. We devise an extensive sample of manifestly bad
astrometric solutions, with parallax that is negative at > 4.5 sigma; and a
corresponding sample of presumably good solutions, including sources in HEALPix
pixels on the sky that do not contain such negative parallaxes, and sources
that fall on the main sequence in a color-absolute magnitude diagram. We then
train a neural network that uses 17 pertinent Gaia catalog entries and
information about nearby sources to discriminate between these two samples,
captured in a single 'astrometric fidelity' parameter. A diverse set of
verification tests shows that our approach works very cleanly, including for
sources with positive parallaxes. The main limitations of our approach are in
the very low-SNR and the crowded regime. Our astrometric fidelities for all of
eDR3 can be queried via the Virtual Observatory, our code and data are public.
