A wide variety of astrophysical and cosmological sources are expected to contribute to a stochasticgravitational-wave background. Following the observations of GW150914 and GW151226, the rate andmass of coalescing binary black holes appear to be greater than many previous expectations. As a result, thestochastic background from unresolved compact binary coalescences is expected to be particularly loud.We perform a search for the isotropic stochastic gravitational-wave background using data from AdvancedLaser Interferometer Gravitational Wave Observatory’s (aLIGO) first observing run. The data display noevidence of a stochastic gravitational-wave signal. We constrain the dimensionless energy density ofgravitational waves to be Ω0 < 1.7 × 10−7 with 95% confidence, assuming a flat energy density spectrumin the most sensitive part of the LIGO band (20–86 Hz). This is a factor of ∼33 times more sensitive thanprevious measurements. We also constrain arbitrary power-law spectra. Finally, we investigate theimplications of this search for the background of binary black holes using an astrophysical model for thebackground.