  We present the first search for gravitational waves from the coalescence of
stellar mass and sub-solar mass black holes with masses between $20 -
100~\mathrm{M}_{\odot}$ and $0.01 - 1~\mathrm{M}_{\odot}~($10 -
10^3$~\mathrm{M}_{J})$, respectively. The observation of a single sub-solar
mass black hole would establish the existence of primordial black holes and a
possible component of dark matter. We search the $\sim 164$ days of public LIGO
data from 2015-2017 when LIGO-Hanford and LIGO-Livingston were simultaneously
observing. We find no significant candidate gravitational-wave signals. Using
this non-detection, we place a $90\%$ upper limit on the rate of
$30-0.01~\mathrm{M}_{\odot}$ and $30-0.1~\mathrm{M}_{\odot}$ mergers at
$<1.2\times10^{6}$ and $<1.6\times10^{4} ~\mathrm{Gpc}^{-3} \mathrm{yr}^{-1}$,
respectively. If we consider binary formation through direct gravitational-wave
braking, this kind of merger would be exceedingly rare if only the lighter
black hole were primordial in origin
($<10^{-4}~\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$). If both black holes are
primordial in origin, we constrain the contribution of $1
(0.1)~\mathrm{M}_{\odot}$ black holes to dark matter to $< 0.3 (3)\%$.
