Very high-energy gamma rays (VHE, E greater than or similar to 100 GeV) propagating over cosmological distances can interact with the low-energy photons of the extragalactic background light (EBL) and produce electron-positron pairs. The transparency of the Universe to VHE gamma rays is then directly related to the spectral energy distribution (SED) of the EBL. The observation of features in the VHE energy spectra of extragalactic sources allows the EBL to be measured, which otherwise is very difficult. An EBL model-independent measurement of the EBL SED with the H.E.S.S. array of Cherenkov telescopes is presented. It was obtained by extracting the EBL absorption signal from the reanalysis of high-quality spectra of blazars. From H.E.S.S. data alone the EBL signature is detected at a significance of 9.5 sigma, and the intensity of the EBL obtained in different spectral bands is presented together with the associated gamma-ray horizon.