The anomalous weak dipole moments of the tau 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(-) --> tau(+)tau(-) at energies close to the Z mass are studied using their semileptonic decays to pi, rho, a(1) --> pi2pi(0) or a(1)--> 3pi. The real and imaginary components of both the anomalous weak magnetic dipole moment and the CP-violating anomalous weak electric dipole moment, Re mu(tau), Im mu(tau), Re d(tau) and Im d(tau), 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 mu(tau)\ < 1.14 x10(-3) \ Im mu(tau)\ < 2.65 x 10(-3), \ Re d(tau)\ < 0.91 x 10(-3), and \Im d(tau)\ < 2.01 x 10(-3).