Using the data collected by the L3 experiment at LEP from 1991 to 1995 at energies around the Z mass, a measurement of the weak anomalous magnetic dipole moment, awτ, and of the weak electric dipole moment, dwτ, of the τ lepton is performed. These quantities are obtained from angular distributions in , where h is a π or a p. The results are: Re(awτ)=(0.0±1.6±2.3)×10−3, Im(awτ)=(−1.0±3.6±4.3)×10−3 and Re(dwτ)=(−0.44±0.88±1.33)×10−17e·cm. This is the first direct measurement of awτ.