We report on an all-sky search for continuous gravitational waves in the frequency band 20-2000 Hz and with a frequency time derivative in the range of [-1.0,+0.1]×10-8 Hz/s. Such a signal could be produced by a nearby, spinning and slightly nonaxisymmetric isolated neutron star in our Galaxy. This search uses the LIGO data from the first six months of Advanced LIGO's and Advanced Virgo's third observational run, O3. No periodic gravitational wave signals are observed, and 95% confidence-level (C.L.) frequentist upper limits are placed on their strengths. The lowest upper limits on worst-case (linearly polarized) strain amplitude h0 are ∼1.7×10-25 near 200 Hz. For a circularly polarized source (most favorable orientation), the lowest upper limits are ∼6.3×10-26. These strict frequentist upper limits refer to all sky locations and the entire range of frequency derivative values. For a population-averaged ensemble of sky locations and stellar orientations, the lowest 95% C.L. upper limits on the strain amplitude are ∼1.4×10-25. These upper limits improve upon our previously published all-sky results, with the greatest improvement (factor of ∼2) seen at higher frequencies, in part because quantum squeezing has dramatically improved the detector noise level relative to the second observational run, O2. These limits are the most constraining to date over most of the parameter space searched.