We present the first Advanced LIGO and Advanced Virgo search for ultracompact binary systems with component masses between 0.2 \(M_\odot\) - 1.0 \(M_\odot\) using data taken between September 12, 2015 and January 19, 2016. We find no viable gravitational wave candidates. Our null result constrains the coalescence rate of monochromatic (delta function) distributions of non-spinning (0.2 \(M_\odot\), 0.2 \(M_\odot\)) ultracompact binaries to be less than \(1.0 \times 10^6 \text{Gpc}^{-3} \text{yr}^{-1}\) and the coalescence rate of a similar distribution of (1.0 \(M_\odot\), 1.0 \(M_\odot\)) ultracompact binaries to be less than \(1.9 \times 10^4 \text{Gpc}^{-3} \text{yr}^{-1}\) (at 90 percent confidence). Neither black holes nor neutron stars are expected to form below ~ 1 solar mass through conventional stellar evolution, though it has been proposed that similarly low mass black holes could be formed primordially through density fluctuations in the early universe. Under a particular primordial black hole binary formation scenario, we constrain monochromatic primordial black hole populations of 0.2 \(M_\odot\) to be less than \(33\%\) of the total dark matter density and monochromatic populations of 1.0 \(M_\odot\) to be less than \(5\%\) of the dark matter density. The latter strengthens the presently placed bounds from micro-lensing surveys of MAssive Compact Halo Objects (MACHOs) provided by the MACHO and EROS collaborations.