Re-observations with the H.E.S.S. telescope array of the very-high-energy (VHE) source HESS J1018-589 A coincident with the Fermi-LAT $gamma$-ray binary 1FGL J1018.6-5856 have resulted in a source detection significance of more than 9$sigma$, and the detection of variability ($chi^2$/$ u$ of 238.3/155) in the emitted $gamma$-ray flux. This variability confirms the association of HESS J1018-589 A with the high-energy $gamma$-ray binary detected by Fermi-LAT, and also confirms the point-like source as a new very-high-energy binary system. The spectrum of HESS J1018-589 A is best fit with a power-law function with photon index $Gamma = 2.20 pm 0.14_{rm stat} pm 0.2_{rm sys}$. Emission is detected up to ~20 TeV. The mean differential flux level is $(2.9 pm 0.4)times10^{-13}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 TeV, equivalent to ~1% of the flux from the Crab Nebula at the same energy. Variability is clearly detected in the night-by-night lightcurve. When folded on the orbital period of 16.58 days, the rebinned lightcurve peaks in phase with the observed X-ray and high-energy phaseograms. The fit of the H.E.S.S. phaseogram to a constant flux provides evidence of periodicity at the level of 3$sigma$. The shape of the VHE phaseogram and measured spectrum suggest a low inclination, low eccentricity system with a modest impact from VHE $gamma$-ray absorption due to pair production ($tau$ < 1 at 300 GeV).
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