The spectral function of the vectorcurrent and the axial-vector current have been measured in hadronic tau decays using the OPAL detector at LEP. Within the framework of the Operator Product Expansion a simultaneous determination of the strong coupling constant alpha_s and several non-perturbative parameters has been performed. The Contour Improved Fixed Order Perturbation Theory gives alpha_s(m_tau^2) = 0.348 pm 0.009_exp pm 0.019_theo at the tau-mass scale and alpha_s(m_Z^2) = 0.1219 pm 0.0010_exp pm 0.0017_theo at the Z-mass scale. The vector spectral function can be used to improve the predictions of the hadronic vacuum polarization corrections to (g-2) of the muon over those derived from e+e- data alone.