We report the discovery of an unidentified, extended source of very-high-energy gamma-ray emission, VER J2019+407, within the radio shell of the supernova remnant SNR G78.2+2.1, using 21.4 hr 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 $0{.\!\!^\circ }23 \pm {0{.\!\!^\circ }03_{\rm stat}} {}^{+0{.\!\!^\circ }04}_{-0{.\!\!^\circ }02{\rm sys}}$ and its spectrum is well-characterized by a differential power law (dN/dE = N 0 × (E/TeV)–Γ) with a photon index of Γ = 2.37 ± 0.14stat ± 0.20sys and a flux normalization of N 0 = 1.5 ± 0.2stat ± 0.4sys × 10–12 photon TeV–1 cm–2 s–1. This yields an integral flux of 5.2 ± 0.8stat ± 1.4sys × 10–12 photon 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 SNR shock.