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The extragalactic background light (EBL) is of fundamental importance both for understanding the entire process of galaxy evolution and for γ-ray astronomy, but the overall spectrum of the EBL between 0.1 and 1000 μm has never been determined directly from galaxy spectral energy distribution (SED) observations over a wide redshift range. The evolving, overall spectrum of the EBL is derived here utilizing a novel method based on observations only. This is achieved from the observed evolution of the rest-frame K-band galaxy luminosity function up to redshift 4, combined with a determination of galaxy-SED-type fractions. These are based on fitting Spitzer Wide-Area Infrared Extragalactic Survey (SWIRE) templates to a multiwavelength sample of about 6000 galaxies in the redshift range from 0.2 to 1 from the All-wavelength Extended Groth Strip International Survey (AEGIS). The changing fractions of quiescent galaxies, star-forming galaxies, starburst galaxies and active galactic nucleus (AGN) galaxies in that redshift range are estimated, and two alternative extrapolations of SED types to higher redshifts are considered. This allows calculation of the evolution of the luminosity densities from the ultraviolet (UV) to the infrared (IR), the evolving star formation rate density of the Universe, the evolving contribution to the bolometric EBL from the different galaxy populations including AGN galaxies and the buildup of the EBL. Our EBL calculations are compared with those from a semi-analytic model, another observationally based model and observational data. The EBL uncertainties in our modelling based directly on the data are quantified, and their consequences for attenuation of very-high-energy γ-rays due to pair production on the EBL are discussed. It is concluded that the EBL is well constrained from the UV to the mid-IR, but independent efforts from IR and γ-ray astronomy are needed in order to reduce the uncertainties in the far-IR.