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, a w τ , and of the weak electric dipole moment, d w τ , of the τ lepton is performed. These quantities are obtained from angular distributions in e + e - ->τ + τ - ->h + τ h - ν τ , where h is a π or a ρ. The results are: Re(a w τ )=(0.0+/-1.6+/-2.3)x10 - 3 , Im(a w τ )=(-1.0+/-3.6+/-4.3)x10 - 3 and Re(d w τ )=(-0.44+/-0.88+/-1.33)x10 - 1 7 e.cm. This is the first direct measurement of a w τ .