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] x 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 01. 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 h 0 is similar to 4 x 10(-25) near 170 Hz, while at the high end of our frequency range, we achieve a worst-case upper limit of 1.3 x 10(-24) . For a circularly polarized source (most favorable orientation), the smallest upper limit obtained is similar to 1.5 x 10(-25)