We present a new measurement of the virtual photon proton asymmetry A(1)(P) from deep inelastic scattering of polarized muons on polarized protons in the kinematic range 0.0008 < x < 0.7 and 0.2 < Q(2) < 100 GeV2. With this, the statistical uncertainty of our measurement has improved by a factor of 2 compared to our previous measurements. The spin-dependent structure function g(1)(P) is determined for the data with Q(2) > 1 GeV2. A perturbative QCD evolution in next-to-leading order is used to determine g(1)(P)(x) as a constant Q(2). At Q(2) = 10 GeV2 we find, in the measured range, integral(0.003)(0.7)g(1)(P)(x)dx = 0.139 +/- 0.006(stat) +/- 0.008(syst)+/- 0.006(evol). The value of the first moment Gamma(1)(P) = integral(0)(1)g(1)(P)(x)dx of g(1)(P) depends on the approach used to describe the behaviour of g(1)(P) at low x. We find that the Ellis-Jaffe sum rule is violated. With our published result for Gamma(1)(d) we confirm the Bjorken sum rule with an accuracy of approximate to 15% at the one standard deviation level. (C) 1997 Elsevier Science B.V.