Spectral lines are among the most powerful signatures for dark matter (DM) annihilation searches in very-high-energy gamma rays. The central region of the Milky Way halo is one of the most promising targets given its large amount of DM and proximity to Earth. We report on a search for a monoenergetic spectral line from self-annihilations of DM particles in the energy range from 300 GeV to 70 TeV using a two-dimensional maximum likelihood method taking advantage of both the spectral and spatial features of the signal versus background. The analysis makes use of Galactic center observations accumulated over ten years (2004-2014) with the H.E.S.S. array of ground-based Cherenkov telescopes. No significant gamma-ray excess above the background is found. We derive upper limits on the annihilation cross section (sigma v) for monoenergetic DM lines at the level of 4 x 10(-28) cm(3) s(-1) at 1 TeV, assuming an Einasto DM profile for the Milky Way halo. For a DM mass of 1 TeV, they improve over the previous ones by a factor of 6. The present constraints are the strongest obtained so far for DM particles in the mass range 300 GeV-70 TeV. Ground-based gamma-ray observations have reached sufficient sensitivity to explore relevant velocity-averaged cross sections for DM annihilation into two gamma-ray photons at the level expected from the thermal relic density for TeV DM particles.