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 ($\chi^2$/$\nu$ 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).
