Neutron diffraction results on single crystals of CsCuCl3 in high magnetic fields are reported. The fields were applied parallel (up to about 6.5 T) and perpendicular (up to 8.5 T) to the c axis of the hexagonal crystal. For the parallel field a spin-flop-like change in the magnetic structure at fields above 5.6 T was observed close below the Neel temperature. The magnetic phase diagram has been determined in the temperature range between 9.6 K and TN=10.675 K. In the perpendicular field the magnetic satellite reflections, originating from the helical magnetic spin structure of the Cu ions, shift their positions in rising field, indicating a stretching out of the helices. The results are interpreted in the light of recent theoretical predictions about the transitions being triggered by quantum or thermal fluctuations. In addition to determining the spin structures, we find that the critical exponent beta is close to 0.25 in the low-field phase and probably close to 0.125 in the high-field phase.