Context : 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 .

Aims : 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 ) .

Methods : 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 .

Results : 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 \Gamma _ { VHE } = 3.45 \pm 0.25 _ { stat } \pm 0.20 _ { syst } .
The integral flux above 200 GeV is ( 1.88 \pm 0.20 _ { stat } \pm 0.38 _ { syst } ) \times 10 ^ { -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 0.2 _ { stat } ) \times 10 ^ { -9 } \mathrm { erg } \mathrm { cm } ^ { -2 } \mathrm { s } ^ { -1 } , and a spectrum well described by a power-law function with a photon index \Gamma _ { HE } = 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 } \mathrm { erg } \mathrm { cm } ^ { -2 } \mathrm { s } ^ { -1 } , and a steep spectrum \Gamma _ { X } = ( 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 .

Conclusions : Although the GeV-TeV observations do not provide better constraints on the EBL than previously obtained , they confirm a low density of the EBL , close to the lower limits from galaxy counts .
The absorption-corrected HE and VHE \gamma -ray spectra are both hard and have similar spectral indices ( \approx 1.86 ) , indicating no significant change of slope between the HE and VHE \gamma -ray bands , and locating the \gamma -ray peak in the SED above 1-2 TeV .
As for other TeV BL Lac objects with the \gamma -ray peak at such high energies and a large separation between the two SED humps , this average broad-band SED represents a challenge for simple one-zone synchrotron self-Compton models , requiring a high Doppler factor and very low B-field .