  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.
