  We present a search for gravitational waves from the coalescence of sub-solar
mass black hole binaries using data from the first half of Advanced LIGO and
Virgo's third observing run. The observation of a sub-solar mass black hole
merger may be an indication of primordial origin; primordial black holes may
contribute to the dark matter distribution. We search for black hole mergers
where the primary mass is $0.1-7 M_{\odot}$ and the secondary mass is $0.1-1
M_{\odot}$. A variety of models predict the production and coalescence of
binaries containing primordial black holes; some involve dynamical assembly
which may allow for residual eccentricity to be observed. For component masses
$>0.5 M_{\odot}$, we also search for sources in eccentric orbits, measured at a
reference gravitational-wave frequency of 10 Hz, up to $e_{10}\sim 0.3$. We
find no convincing candidates and place new upper limits on the rate of
primordial black hole mergers. The merger rate of 0.5-0.5 (1.0-1.0)~$M_{\odot}$
sources is $<7100~(1200)$ Gpc$^{-3}$yr$^{-1}$. Our limits are $\sim3-4$ times
more constraining than prior analyses. Finally, we demonstrate how our limits
can be used to constrain arbitrary models of the primordial black hole mass
distribution and merger rate.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology