We report on the measurement of the beam spin asymmetry in the deeply virtual Compton scattering off He-4 using the CEBAF Large Acceptance Spectrometer (CLAS) at Jefferson Lab using a 6 GeV longitudinally polarized electron beam incident on a pressurized He-4 gaseous target. We detail the method used to ensure the exclusivity of the measured reactions, in particular the upgrade of CLAS with a radial time projection chamber to detect the low-energy recoiling He-4 nuclei and an inner calorimeter to extend the photon detection acceptance at forward angles. Our results confirm the theoretically predicted enhancement of the coherent (e(4)He -> e'4He'gamma') beam spin asymmetries compared to those observed on the free proton, while the incoherent (e(4)He -> e'p'gamma'X') asymmetries exhibit a 30% suppression. From the coherent data, we were able to extract, in a model-independent way, the real and imaginary parts of the only He-4 Compton form factor, H-A, leading the way toward 3D imaging of the partonic structure of nuclei.