We used existing data from the New Horizons LORRI camera to measure the cosmic optical background (COB). After removing foregrounds and scattered light, we detect the COB signal at ≥3.8-sigma significance. When we subtract off the additional contribution expected for galaxies below the LORRI detection limit and down to V=30, we find a residual “diffuse COB” (dCOB) signal that is in the range 8.8 ± 4.9 (1.8 stat., 4.5 sys.) nW m-2 sr-1 to 11.9 ± 4.6 (1.8 stat., 4.2 sys.) nW m-2 sr-1. The surface brightness values corresponding to these dCOB intensities are 27.3 (+0.9, -0.5) mag arcsec-2 and 27.0 (+0.5, -0.4) mag arcsec-2, respectively, in the AB system computed at the LORRI pivot wavelength of 0.608 micron. The total error in our COB and dCOB measurements include the uncertainties in the determination of the faint-end slope of the galaxy counts and the effects of cosmic variance. To attribute this diffuse COB signal to undetected faint galaxies requires either that the faint-end slope of the galaxy count - magnitude relation would need to steepen markedly at V ≥ 24 or that the corrected galaxy counts from a variety of existing deep surveys are missing half the galaxies with V ≤ 30. Our diffuse COB signal in the optical passband of LORRI (0.4 - 0.9 micron) is consistent with constraints measured in the range 0.8 - 1.8 microns using NIR spectra of the sky obtained by the CIBER rocket instrument. One origin for such a diffuse COB could be faint compact objects at z ≤ 0.1. A key argument for sources at low redshift is accommodating the very high energy gamma-ray opacity limits on COB components at high redshift.