High-spin states in 135Nd were populated with the 110Pd(30Si,5n)135Nd reaction at a 30Si bombarding energy of 133 MeV. Two DeltaI=1 bands with close excitation energies and the same parity were observed. These bands are directly linked by DeltaI=1 and DeltaI=2 transitions. The chiral nature of these two bands is confirmed by comparison with three-dimensional tilted axis cranking calculations. This is the first observation of a three-quasiparticle chiral structure and establishes the primarily geometric nature of this phenomenon.