  Using recently developed methods for the evaluation of five-loop amplitudes
in perturbative QCD, corrections of order alpha_s^4 for the cross section of
electron-positron annihilation into hadrons and for the decay rates of the
Z-boson and the tau-lepton into hadrons are evaluated. The new terms lead to a
significant stabilization of the perturbative series, to a reduction of the
theory uncertainly in the strong coupling constant alpha_s, as extracted from
these measurements, and to a small shift of the central value, moving two
central values closer together. The agreement between two values of alpha_s
measured at vastly different energies constitutes a striking test of asymptotic
freedom. Combining the results from Z and tau decays we find
alpha_s(M_Z)=0.1198 \pm 0.0015 as one of the most precise and presently only
result for the strong coupling constant in order alpha_s^4.
