We obtained New Horizons LORRI images to measure the cosmic optical background (COB) intensity integrated over 0.4 μm ≲ λ ≲ 0.9 μm. The survey comprises 16 high-Galactic-latitude fields selected to minimize scattered diffuse Galactic light (DGL) from the Milky Way, as well as scattered light from bright stars. This work supersedes an earlier analysis based on observations of one of the present fields. Isolating the COB contribution to the raw total sky levels measured in the fields requires subtracting the remaining scattered light from bright stars and galaxies, intensity from faint stars within the fields fainter than the photometric detection limit, and the DGL foreground. DGL is estimated from 350 μm and 550 μm intensities measured by the Planck High Frequency Instrument, using a new self-calibrated indicator based on the 16 fields augmented with eight additional DGL calibration fields obtained as part of the survey. The survey yields a highly significant detection (6.8σ) of the COB at 11.16 ± 1.65 (1.47 sys, 0.75 ran) nW m−2 sr−1 at the LORRI pivot wavelength of 0.608 μm. The estimated integrated intensity from background galaxies, 8.17 ± 1.18 nW m−2 sr−1, can account for the great majority of this signal. The rest of the COB signal, 2.99 ± 2.03 (1.75 sys, 1.03 ran) nW m−2 sr−1, is formally classified as anomalous intensity but is not significantly different from zero. The simplest interpretation is that the COB is completely due to galaxies.