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, a1 → π2π0 or a1 → 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 μτ, Re dτ 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 μτ| < 1.14 × 10-3, |Im μτ| < 2.65 × 10-3, |Re dτ| < 0.91 × 10-3, and |Im dτ| < 2.01 × 10-3.