The parity-violating cross-section asymmetry in the elastic scattering of polarized electrons from unpolarized protons has been measured at a four-momentum transfer squared Q(2) = 0.624 GeV2 and beam energy E-b = 3.48 GeV to be A(PV) = -23.80 +/- 0.78(stat) +/- 0.36(syst) parts per million. This result is consistent with zero contribution of strange quarks to the combination of electric and magnetic form factors G(E)(s) + 0.517G(M)(s) = 0.003 +/- 0.010(stat) +/- 0.004(syst) +/- 0.009(ff), where the third error is due to the limits of precision on the electromagnetic form factors and radiative corrections. With this measurement, the world data on strange contributions to nucleon form factors are seen to be consistent with zero and not more than a few percent of the proton form factors.