For τ leptons produced in e+e− → τ+τ− interactions there are, in addition to the longitudinal spin correlations, two independent transverse spin correlations associated with the transverse (within the production plane) and normal (to the production plane) polarization components. A measurement of the transverse-transverse and transverse-normal τ spin correlations in the decay Z → τ+τ−, CTT and CTN, is presented based on the aplanarity angle of the decay products of both τ leptons. Using 80 pb−1 of data collected by ALEPH on the peak of the Z resonance, the results are CTT = 1.06±0.13 (stat) ±0.05(syst), and CTN = 0.08±0.13(stat)±0.04(syst). These values are in agreement with the Standard Model predictions, CTT = 0.99 and CTN = −0.01.