  We describe a search for gravitational waves from compact binaries with at
least one component with mass 0.2 $M_\odot$ -- $1.0 M_\odot$ and mass ratio $q
\geq 0.1$ in Advanced LIGO and Advanced Virgo data collected between 1 November
2019, 15:00 UTC and 27 March 2020, 17:00 UTC. No signals were detected. The
most significant candidate has a false alarm rate of 0.2 $\mathrm{yr}^{-1}$. We
estimate the sensitivity of our search over the entirety of Advanced LIGO's and
Advanced Virgo's third observing run, and present the most stringent limits to
date on the merger rate of binary black holes with at least one subsolar-mass
component. We use the upper limits to constrain two fiducial scenarios that
could produce subsolar-mass black holes: primordial black holes (PBH) and a
model of dissipative dark matter. The PBH model uses recent prescriptions for
the merger rate of PBH binaries that include a rate suppression factor to
effectively account for PBH early binary disruptions. If the PBHs are
monochromatically distributed, we can exclude a dark matter fraction in PBHs
$f_\mathrm{PBH} \gtrsim 0.6$ (at 90% confidence) in the probed subsolar-mass
range. However, if we allow for broad PBH mass distributions we are unable to
rule out $f_\mathrm{PBH} = 1$. For the dissipative model, where the dark matter
has chemistry that allows a small fraction to cool and collapse into black
holes, we find an upper bound $f_{\mathrm{DBH}} < 10^{-5}$ on the fraction of
atomic dark matter collapsed into black holes.
