We report the most precise measurement to date of a parity-violating asymmetry in elastic electron-proton scattering. The measurement was carried out with a beam energy of 3.03 GeV and a scattering angle <theta_lab>=6 degrees, with the result A_PV = -1.14 +/- 0.24 (stat) +/- 0.06 (syst) parts per million. From this we extract, at Q^2 = 0.099 GeV^2, the strange form factor combination G_E^s + 0.080 G_M^s = 0.030 +/- 0.025 (stat) +/- 0.006 (syst) +/- 0.012 (FF) where the first two errors are experimental and the last error is due to the uncertainty in the neutron electromagnetic form factor. This result significantly improves current knowledge of G_E^s and G_M^s at Q^2 ~0.1 GeV^2. A consistent picture emerges when several measurements at about the same Q^2 value are combined: G_E^s is consistent with zero while G_M^s prefers positive values though G_E^s=G_M^s=0 is compatible with the data at 95% C.L.
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