Neutron diffraction investigations have been performed to study the magnetization process of CsCuCl₃ with the magnetic field aligned within the ab-plane. In zero field the stacked, triangular-lattice antiferromagnet (TLA) CsCuCl₃ has a helical structure incommensurate in the chain direction normal to the ab-plane. The magnetic phase diagram was investigated from 2 K up to T_N in fields up to 17 T. The phase line for the expected incommensurate-commensurate (IC-C) phase transition could be determined throughout the whole phase diagram. At low temperature the IC-C transition is roughly at half the saturation field H_S. The neutron diffraction patterns were found to be well described by a sinusoidally modulated spiral in fields up to H_S/3. The initial increase of the scattering intensity in rising field indicates a continuous reduction of the spin frustration on the triangular lattice. Between H_S/3 and H_S/2 a new phase occurs where the spiral vector has a plateau in its field dependence. Close to the IC-C transition a growing asymmetry of magnetic satellite-peak intensities indicates domain effects which are related to the lifting of the chiral degeneracy in the ab-plane in rising field. The phase diagram obtained has some similarities with those calculated for stacked TLAs by considering the effects of quantum and thermal fluctuations.