The class of 1-D ferromagnetic chain systems AFeCl3 (A=Rb,Tl,Cs) are of special interest in the group of 1-D magnets of the type ABX3. This is because of the strong single-ion-anisotropy D, which favors a nonmagnetic singlet-ground state. If however the exchange interaction J is large enough, with d=D/J<1 the system has a magnetic ground state with easy plane anisotropy. While RbFeCl3 has a magnetic ground state, CsFeCl3 has not; it is however possible to induce a magnetic moment through the Zeeman splitting of the first excited state. We have therefore studied CsFeCl3 by neutron scattering experiments in external magnetic fields, H̄∥c̄. Such experiments yield a strong quasi-elastic scattering around Q=(0.294,0.353,0) and Q=(0.353,0.294,0) at H≥4 T, due to the dynamical behavior of the correlated clusters of spins in the easy plane. At higher fields or lower temperatures true elastic scattering occurs at incommensurable positions with Q̄=(1/3−x,1/3−x,0). Further increase of fields or decrease of temperature leads to discrete changes of x, which finally lock in at the commensurate 120 spin structure. We explain the incommensurate phases as a result of the competition between dipole- and exchange-interaction. In addition we performed inelastic measurements in fields up to 5.4 T, which cannot be explained by spinwave theory. These inelastic results are discussed on the basis of the exciton model using the RPA decoupling.