Aims. Detection of 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 HESS Galactic plane survey to longitudes > 30 degrees. Results. HESS J1908+063 is detected at a significance level of 10.9 sigma with an integral flux above 1 TeV of (3.76 +/- 0.29 (stat) +/- 0.75(sys)) x 10(-12) ph cm(-2) s(-1), and a spectral photon index Gamma = 2.10 +/- 0.07 (stat) +/- 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.