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  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 ± 0.14_stat ± 0.20_sys and a flux normalization of N ₀ = 1.5 ± 0.2_stat ± 0.4_sys × 10^–12 photon TeV^–1 cm^–2 s^–1. This yields an integral flux of 5.2 ± 0.8_stat ± 1.4_sys × 10^–12 photon cm^–2 s^–1above 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.