The photon asymmetry in the reaction 1H(--> gamma,pi(0))(1)H close to threshold has been measured for the first time with the photon spectrometer TAPS using linearly polarized photons from the tagged-photon facility at the Mainz Microtron MAMI. The total and differential cross sections were also measured simultaneously with the photon asymmetry. This allowed determination of the S-wave and all three P-wave amplitudes. The values obtained at threshold are E(0+) = [-1.33+/-0.08(stat)+/-0.03(syst)] x 10(-3)/m(pi(+)), P(1) = [9.47 +/- 0.08(stat) +/- 0.29(syst)] x 10(-3)q/m(2)(pi(+)), P(2) = [-9.46 +/- 0.1(stat) +/- 0.29(syst)] x 10(-3)q/m(2)(pi(+)), and P(3) = [11.48 +/- 0.06(stat) +/- 0.35(syst)] x 10(-3)q/m(2)(pi(+)).