The population of high-energy and very high-energy γ-ray sources, detected with EGRET and the new generation of ground-based Cherenkov telescopes, forms a reduced but physically important sample. Most of these sources are extragalactic (e.g. blazars), while among the galactic ones there are pulsars and supernova remnants. The microquasar LS 5039, previously proposed to be associated with an EGRET source by Paredes et al., has recently been detected at TeV energies, confirming that microquasars should be regarded as a class of high-energy γ-ray sources. To model and understand how the energetic photons are produced and escape from LS 5039, it is crucial to unveil the nature of the compact object, which remains unknown. Here, we present new intermediate-dispersion spectroscopy of this source, which, combined with values reported in the literature, provides an orbital period of Porb= 3.90603 +/- 0.00017 d, a mass function f(M) = 0.0053 +/- 0.0009Msolar and an eccentricity e= 0.35 +/- 0.04. Atmosphere model fitting to the spectrum of the optical companion, together with our new distance estimate of d= 2.5 +/- 0.1 kpc, yields RO= 9.3+0.7-0.6Rsolar, log(LO/Lsolar) = 5.26 +/- 0.06 and MO= 22.9+3.4-2.9Msolar. These, combined with our dynamical solution and the assumption of pseudo-synchronization, yield an inclination and a compact object mass MX= 3.7+1.3-1.0Msolar. This is above neutron star masses for most of the standard equations of state and, therefore, we propose that the compact object in LS 5039 is a black hole. We finally discuss the implications of our orbital solution and new parameters of the binary system on the CNO products, the accretion/ejection energetic balance, the supernova explosion scenario and the behaviour of the very high-energy γ-ray emission with the new orbital period.