An annihilation signal of dark matter is searched for from the central region of the Milky Way. Data acquired in dedicated ON/OFF observations of the Galactic center region with H.E.S.S. are analyzed for this purpose. No significant signal is found in a total of $sim 9$ h of ON/OFF observations. Upper limits on the velocity averaged cross section, $<sigma v >$, for the annihilation of dark matter particles with masses in the range of $sim 300$ GeV to $sim 10$ TeV are derived. In contrast to previous constraints derived from observations of the Galactic center region, the constraints that are derived here apply also under the assumption of a central core of constant dark matter density around the center of the Galaxy. Values of $<sigma v >$ that are larger than $3cdot 10^{-24}:mathrm{cm^3/s}$ are excluded for dark matter particles with masses between $sim 1$ and $sim 4$ TeV at 95% CL if the radius of the central dark matter density core does not exceed $500$ pc. This is the strongest constraint that is derived on $<sigma v>$ for annihilating TeV mass dark matter without the assumption of a centrally cusped dark matter density distribution in the search region.
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