Multiwavelength observations of the high-frequency-peaked blazar 1ES 2344+514 were performed from 2007 October to 2008 January. The campaign represents the first contemporaneous data on the object at very high energy (VHE, E >100 GeV) Delta *g-ray, X-ray, and UV energies. Observations with VERITAS in VHE Delta *g-rays yield a strong detection of 20 Delta *s with 633 excess events in a total exposure of 18.1 hr live time. A strong VHE Delta *g-ray flare on 2007 December 7 is measured at F(>300 GeV) = (6.76 ? 0.62) X 10--11 photons cm--2 s--1, corresponding to 48% of the Crab Nebula flux. Excluding this flaring episode, nightly variability at lower fluxes is observed with a time-averaged mean of F(>300 GeV) = (1.06 ? 0.09) X 10--11 photons cm--2 s--1 (7.6% of the Crab Nebula flux). The differential photon spectrum between 390 GeV and 8.3 TeV for the time-averaged observations excluding 2007 December 7 is well described by a power law with a photon index of Delta *G = 2.78 ? 0.09stat ? 0.15syst. On the flaring night of 2007 December 7 the measured VHE Delta *g-ray photon index was Delta *G = 2.43 ? 0.22stat ? 0.15syst. Over the full period of VERITAS observations contemporaneous X-ray and UV data were taken with Swift and RXTE. The measured 2-10 keV flux ranged by a factor of ~7 during the campaign. On 2007 December 8 the highest ever observed X-ray flux from 1ES 2344+514 was measured by Swift X-ray Telescope at a flux of F(2-10 keV) = (6.28 ? 0.31) X 10--11 erg cm--2 s--1. Evidence for a correlation between the X-ray flux and VHE Delta *g-ray flux on nightly timescales is indicated with a Pearson correlation coefficient of r = 0.60 ? 0.11. Contemporaneous spectral energy distributions (SEDs) of 1ES 2344+514 are presented for two distinct flux states. A one-zone synchrotron self-Compton (SSC) model describes both SEDs using parameters consistent with previous SSC modeling of 1ES 2344+514 from non-contemporaneous observations.