The Goddard High Resolution Spectrograph of the Hubble Space Telescope has been used to obtain spectra of three halo dwarfs at a resolution of 25,000 for a region centered on the B I resonance lines near 2497 angstrom. The weaker component of the B i doublet is detected in all three stars, but the stronger component is irretrievably blended with strong Fe lines. The stars observed are HD 19445, 140283, and 201891, with metallicities of [Fe/H] = - 2.15, - 2.64, and - 1.06 and B abundances of log epsilon(B) = 0.4, - 0.1, and 1.7, respectively, on the scale where log epsilon(H) = 12.0 and log epsilon(Fe) = 7 for the Sun. Using recent determinations of the Be abundance in HD 140283, we find B/Be = 10(-4)+5 for this star and infer similar ratios for HD 19445 and HD 201891. This ratio is equal to the minimum value of 10 expected from synthesis of B and Be by high-energy cosmic-ray spallation reactions in the interstellar medium. It is shown that the accompanying synthesis of Li by a on a fusion reactions is probably a minor contributor to the observed "primordial" Li of halo stars. It is noted that the observed constant ratios of B/O and Be/O are expected if the principal channel of synthesis involves cosmic-ray CNO nuclei from supernovae colliding with interstellar protons. The relations are also the expected result if cosmic rays in the halo were provided by a source, possibly external to the Galaxy, other than halo stars.