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We report on the discovery of high-energy (HE; E > 0.1 GeV) and very high energy (VHE; E > 100 GeV) gamma-ray emission from the high-frequency-peaked BL Lac object RBS 0413. VERITAS, a ground-based gamma-ray observatory, detected VHE. rays from RBS 0413 with a statistical significance of 5.5 standard deviations (sigma) and a gamma-ray flux of (1.5 +/- 0.6(stat) +/- 0.7(syst)) x 10(-8) photons m(-2) s(-1) (similar to 1% of the Crab Nebula flux) above 250 GeV. The observed spectrum can be described by a power law with a photon index of 3.18 +/- 0.68(stat) +/- 0.30(syst). Contemporaneous observations with the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope detected HE gamma rays from RBS 0413 with a (stat)istical significance of more than 9 sigma, a power-law photon index of 1.57 +/- 0.12(stat-0.12sys')(+0.11) and a gamma-ray flux between 300 MeV and 300 GeV of (1.64 +/- 0.43(stat-0.22sys)(+ 0.31)) x 10(-5) photons m(-2) s(-1). We present the results from Fermi-LAT and VERITAS, including a spectral energy distribution modeling of the gamma-ray, quasi-simultaneous X-ray (Swift-XRT), ultraviolet (Swift-UVOT), and R-band optical (MDM) data. We find that, if conditions close to equipartition are required, both the combined synchrotron self-Compton/external-Compton and the lepto-hadronic models are preferred over a pure synchrotron self-Compton model.