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 ∼9 h of on-off observations. Upper limits on the velocity averaged cross section, ⟨σv⟩, for the annihilation of dark matter particles with masses in the range of ∼300 GeV to ∼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 ⟨σv⟩ that are larger than 3×10^{-24} cm^{3}/s are excluded for dark matter particles with masses between ∼1 and ∼4 TeV at 95% C.L. if the radius of the central dark matter density core does not exceed 500 pc. This is the strongest constraint that is derived on ⟨σv⟩ for annihilating TeV mass dark matter without the assumption of a centrally cusped dark matter density distribution in the search region.