  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.
