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The inner region of the Milky Way halo harbors a large amount of dark matter (DM). Given its proximity, it is one of the most promising targets to look for DM. We report on a search for the annihilations of DM particles using gamma-ray observations towards the inner 300 pc of the Milky Way, with the H.E.S.S. array of ground-based Cherenkov telescopes. The analysis is based on a 2D maximum likelihood method using Galactic Center (GC) data accumulated by H.E.S.S. over the last 10 years (2004-2014), and does not show any significant gamma-ray signal above background. Assuming Einasto and Navarro-Frenk-White DM density profiles at the GC, we derive upper limits on the annihilation cross section <sigma nu >. These constraints are the strongest obtained so far in the TeV DM mass range and improve upon previous limits by a factor 5. For the Einasto profile, the constraints reach <sigma nu > values of 6 x 10(-26) cm(3) s(-1) in the W+W- channel for a DM particle mass of 1.5 TeV, and 2 x 10(-26) cm(3) s(-1) in the tau(+)tau(-) channel for a 1 TeV mass. For the first time, ground-based gamma-ray observations have reached sufficient sensitivity to probe <sigma nu > values expected from the thermal relic density for TeV DM particles.