The anomalous weak dipole moments of the τ lepton are measured in a data sample collected by ALEPH from 1990 to 1995 corresponding to an integrated luminosity of 155 pb −1 . Tau leptons produced in the reaction e + e − → τ + τ − at energies close to the Z mass are studied using their semileptonic decays to π, ρ, a 1 → π2π 0 or a 1 → 3π. The real and imaginary components of both the anomalous weak magnetic dipole moment and the CP-violating anomalous weak electric dipole moment, Re µ τ , Im µ τ , Re d τ and Im d τ , are measured simultaneously by means of a likelihood fit built from the full differential cross section. No evidence of new physics is found. The following bounds are obtained (95% CL): |Re µ τ | < 1.14 ×10 −3 , |Im µ τ | < 2.65 ×10 −3 , |Re d τ | < 0.91 ×10 −3 , and |Im d τ | < 2.01 × 10 −3 .