Context : Aims : Detection of a \gamma -ray source above 300 GeV is reported , confirming the unidentified source MGRO J1908+06 , discovered by the Milagro collaboration at a median energy of 20 TeV . Methods : The source was observed during 27 h as part of the extension of the H.E.S.S . Galactic plane survey to longitudes > 30 \degr . Results : HESS J1908+063 is detected at a significance level of 10.9 \sigma with an integral flux above 1 TeV of ( 3.76 \pm 0.29 _ { ~ { } stat } \pm 0.75 _ { sys } ) \times 10 ^ { -12 } ph cm ^ { -2 } s ^ { -1 } , and a spectral photon index \Gamma = 2.10 \pm 0.07 _ { ~ { } stat } \pm 0.2 _ { sys } . The positions and fluxes of HESS J1908+063 and MGRO J1908+06 are in good agreement . Possible counterparts at other wavelengths and the origin of the \gamma -ray emission are discussed . The nearby unidentified GeV source , GRO J1908+0556 ( GeV ) which also remains unidentified and the new Fermi pulsar 0FGL J1907.5+0617 , may be connected to the TeV source . Conclusions :