We present the first Advanced LIGO and Advanced Virgo search for ultracompact binary systems with component masses between 0.2 M⊙–1.0 M⊙ 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 nonspinning (0.2 M⊙, 0.2 M⊙) ultracompact binaries to be less than 1.0×106 Gpc−3 yr−1 and the coalescence rate of a similar distribution of (1.0 M⊙, 1.0 M⊙) ultracompact binaries to be less than 1.9×104 Gpc−3 yr−1 (at 90% confidence). Neither black holes nor neutron stars are expected to form below ∼1 M⊙ 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 and contribute to the dark matter density. The interpretation of our constraints in the primordial black hole dark matter paradigm is highly model dependent; however, under a particular primordial black hole binary formation scenario we constrain monochromatic primordial black hole populations of 0.2 M⊙ to be less than 33% of the total dark matter density and monochromatic populations of 1.0 M⊙ to be less than 5% of the dark matter density. The latter strengthens the presently placed bounds from microlensing surveys of massive compact halo objects (MACHOs) provided by the MACHO and EROS Collaborations.