This paper describes a measurement of the Michel parameters, \eta, \rho, \xi, \xi\delta, and the average \nu_{\tau} helicity, h_{\nu_{\tau}}, in tau lepton decays together with the first measurement of the tensor coupling in the weak charged current. The tau+tau- pairs were produced at the LEP e+e- collider at CERN from 1992 through 1995 in the DELPHI detector. Assuming lepton universality in the decays of the tau the measured values of the parameters were: \eta = -0.005 +/- 0.036 +/- 0.037, \rho = 0.775 +/- 0.023 +/- 0.020, \xi = 0.929 +/- 0.070 +/- 0.030, \xi\delta = 0.779 +/- 0.070 +/- 0.028, h_{\nu_{\tau}} = -0.997 +/- 0.027 +/- 0.011. The strength of the tensor coupling was measured to be \kappa^W_\tau = -0.029 +/- 0.036 +/- 0.018. The first error is statistical and the second error is systematic in all cases. The results are consistent with the V-A structure of the weak charged current in decays of the tau lepton.