1ES 0414+009 (z = 0.287) is a distant high-frequency-peaked BL Lac object, and has long been considered a likely emitter of very-high energy (VHE, E>100 GeV) gamma-rays due to its high X-ray and radio flux. Observations in the VHE gamma-ray band and across the electromagnetic spectrum can provide insights into the origin of highly energetic particles present in the source and the radiation processes at work. Because of the distance of the source, the gamma-ray spectrum might provide further limits on the level of the Extragalactic Background Light (EBL). We report observations made between October 2005 and December 2009 with H.E.S.S., an array of four imaging atmospheric Cherenkov telescopes. Observations at high energies (HE, 100 MeV - 100 GeV) with the Fermi-LAT instrument in the first 20 months of its operation are also reported. To complete the multi-wavelength picture, archival UV and X-ray observations with the Swift satellite and optical observations with the ATOM telescope are also used. Based on the observations with H.E.S.S., 1ES 0414+009 is detected for the first time in the VHE band. An excess of 224 events is measured, corresponding to a significance of 7.8 sigma. The photon spectrum of the source is well described by a power law, with photon index of 3.45 pm 0.25stat pm 0.20syst. The integral flux above 200 GeV is (1.88 pm 0.20stat pm 0.38syst) times10-12 cm-2 s-1. Observations with the Fermi-LAT in the first 20 months of operation show a flux between 200 MeV and 100 GeV of (2.3 pm 0.2stat) times 10-9 erg cm-2 s-1, and a spectrum well described by a power-law function with a photon index 1.85 pm 0.18. Swift/XRT observations show an X-ray flux between 2 and 10 keV of (0.8 - 1) times 10-11 erg cm-2 s-1, and a steep spectrum (2.2 - 2.3). Combining X-ray with optical-UV data, a fit with a log-parabolic function locates the synchrotron peak around 0.1 keV. ...
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