4 pages.-- Printed version published on Feb 20, 2008.LS 5039 is one of a handful of X-ray binaries that have been recently detected at high-energy γ-rays, in this case, by the High Energy Stereoscopic System (H.E.S.S.). The nature of this system is unknown: both a black hole and a pulsar have been invoked as possible compact object companions. Here we work with a model of the high-energy phenomenology of the system in which it is assumed that the companion object is a pulsar rotating around an O6.5 V star in a ~3.9 day orbit. The model assumes two different sets of power-law spectral parameters of the interacting primary leptons corresponding to the two orbital phase intervals defined by H.E.S.S. as having different γ-ray spectra and very high energy (VHE) cutoffs. We show that the H.E.S.S. phenomenology is completely explained by this model. We present predictions for photons with lower energies (for E > 1 GeV), subject to test in the forthcoming months with the GLAST satellite. We find that GLAST will be able to judge on this model within 1 year.We acknowledge support by grants MEC-AYA 2006-00530 and CSIC-PIE 200750I029.Peer reviewed