We report the discovery of very-high-energy ( VHE ) \gamma -ray emission of the binary system PSR B1259 - 63 / SS 2883 of a radio pulsar orbiting a massive , luminous Be star in a highly eccentric orbit . The observations around the 2004 periastron passage of the pulsar were performed with the four 13 m Cherenkov telescopes of the H.E.S.S . experiment , recently installed in Namibia and in full operation since December 2003 . Between February and June 2004 , a \gamma -ray signal from the binary system was detected with a total significance above 13 \sigma . The flux was found to vary significantly on timescales of days which makes PSR B1259 - 63 the first variable galactic source of VHE \gamma -rays observed so far . Strong emission signals were observed in pre- and post-periastron phases with a flux minimum around periastron , followed by a gradual flux decrease in the months after . The measured time-averaged energy spectrum above a mean threshold energy of 380 GeV can be fitted by a simple power law F _ { 0 } ( E / 1 TeV ) ^ { - \Gamma } with a photon index \Gamma = 2.7 \pm 0.2 _ { \mathrm { stat } } \pm 0.2 _ { \mathrm { sys } } and flux normalisation F _ { 0 } = ( 1.3 \pm 0.1 _ { \mathrm { stat } } \pm 0.3 _ { \mathrm { sys } } ) \times 10 ^ { -12 } TeV% ^ { -1 } cm ^ { -2 } s ^ { -1 } . This detection of VHE \gamma -rays provides unambiguous evidence for particle acceleration to multi-TeV energies in the binary system . In combination with coeval observations of the X-ray synchrotron emission by the RXTE and INTEGRAL instruments , and assuming the VHE \gamma -ray emission to be produced by the inverse Compton mechanism , the magnetic field strength can be directly estimated to be of the order of 1 G .