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 } _ { \mathrm { s } tat } } { } ^ { +0.04 ^ { \circ } } _ { -0.02 ^ { % \circ } { \mathrm { s } ys } } 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 } _ { \mathrm { s } tat } \pm { 0.20 } _ { \mathrm { s } ys } and a flux normalization of N _ { 0 } = 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 .