  A beam-normal single-spin asymmetry generated in the scattering of
transversely polarized electrons from unpolarized nucleons is an observable
related to the imaginary part of the two-photon exchange process. We report a
2% precision measurement of the beam-normal single-spin asymmetry in elastic
electron-proton scattering with a mean scattering angle of theta_lab = 7.9
degrees and a mean energy of 1.149 GeV. The asymmetry result is B_n = -5.194 +-
0.067 (stat) +- 0.082 (syst) ppm. This is the most precise measurement of this
quantity available to date and therefore provides a stringent test of
two-photon exchange models at far-forward scattering angles (theta_lab -> 0)
where they should be most reliable.
