We report on a new all-sky search for periodic gravitational waves in the frequency band 475–2000 Hz and with a frequency time derivative in the range of [−1.0,+0.1]×10−8  Hz/s. Potential signals could be produced by a nearby spinning and slightly nonaxisymmetric isolated neutron star in our Galaxy. This search uses the data from Advanced LIGO’s first observational run O1. No gravitational-wave signals were observed, and upper limits were placed on their strengths. For completeness, results from the separately published low-frequency search 20–475 Hz are included as well. Our lowest upper limit on worst-case (linearly polarized) strain amplitude h0 is ∼4×10−25 near 170 Hz, while at the high end of our frequency range, we achieve a worst-case upper limit of 1.3×10−24. For a circularly polarized source (most favorable orientation), the smallest upper limit obtained is ∼1.5×10−25.