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New observations of LS 5039, a High Mass X-ray Binary comprising a massivestar and compact object, were carried out with the High Energy StereoscopicSystem of Cherenkov Telescopes (H.E.S.S.) in 2005 at very high energy (VHE)gamma-ray energies. These observations reveal that its flux and energy spectrumare modulated with the 3.9 day orbital period of the binary system. This is thefirst time in gamma-ray astronomy that orbital modulation has been observed,and periodicity clearly established using ground-based gamma-ray detectors. TheVHE gamma-ray emission is largely confined to half of the orbit, peaking aroundthe inferior conjunction epoch of the compact object. For this epoch, there isalso a hardening of the energy spectrum in the energy range between 0.2 TeV anda few TeV. The flux vs. orbital phase profile provides the first clearindication of gamma-ray absorption via pair production within an astrophysicalsource, a process which is expected to occur if the gamma-ray production siteis situated within ~1 AU of the compact object. Moreover the production regionsize must be not significantly greater than the binary separation (~0.15 AU).Notably, these constraints are also considerably smaller than the collimatedoutflows or jets (extending out to ~1000 AU) observed in LS 5039. The spectralhardening could arise from variations with phase in the maximum electronenergies, and/or the dominant VHE gamma-ray production mechanism.Comment: 8 pages, 8 figures, accepted for publication in Astronomy &  Astrophysic

3.9 day orbital modulation in the TeV γ-ray flux and spectrum from the X-ray binary LS 5039