We report the discovery of an unidentified, extended source of very-high-energy (VHE) gamma-ray emission, VER J2019+407, within the radio shell of the supernova remnant SNR G78.2+2.1, using 21.4 hours of data taken by the VERITAS gamma-ray observatory in 2009. These data confirm the preliminary indications of gamma-ray emission previously seen in a two-year (2007-2009) blind survey of the Cygnus region by VERITAS. VER J2019+407, which is detected at a post-trials significance of 7.5 standard deviations in the 2009 data, is localized to the northwestern rim of the remnant in a region of enhanced radio and X-ray emission. It has an intrinsic extent of 0.23^{circ} pm 0.03^{circ} (stat)+0.04^{circ}_{-0.02}^{circ}(sys) and its spectrum is well-characterized by a differential power law (dN/dE = N_0 times (E/TeV)^{-Gamma}) with a photon index of {Gamma} = 2.37 pm 0.14 (stat) pm 0.20 (sys) and a flux normalization of N0 = 1.5 pm 0.2 (stat) pm 0.4(sys) times 10^-12 ph TeV^{-1} cm^{-2} s^{-1}. This yields an integral flux of 5.2 pm 0.8 (stat) pm 1.4 (sys) times 10^-12 ph cm^{-2} s^{-1} above 320 GeV, corresponding to 3.7% of the Crab Nebula flux. We consider the relationship of the TeV gamma-ray emission with the GeV gamma-ray emission seen from SNR G78.2+2.1 as well as that seen from a nearby cocoon of freshly accelerated cosmic rays. Multiple scenarios are considered as possible origins for the TeV gamma-ray emission, including hadronic particle acceleration at the supernova remnant shock.
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