We present observations of the BL Lac object 1ES 0414+009 in the >200 GeV gamma-ray band by the VERITAS array of Cherenkov telescopes. 1ES 0414+009 was observed by VERITAS between January 2008 and February 2011, resulting in 56.2 hours of good quality pointed observations. These observations resulted in a detection of 822 events from the source corresponding to a statistical significance of 6.4 standard deviations (6.4 sigma) above the background. The source flux, showing no evidence for variability, is measured as 5.2 +/- 1.1_stat +/- 2.6_sys * 10^-12 photons cm^-2 s^-1 above 200 GeV, equivalent to approximately 2% of the Crab Nebula flux above this energy. The differential photon spectrum from 230 GeV to 850 GeV is well fit by a power law with an photon index of Gamma 3.4 +/- 0.5_stat +/- 0.3_sys and a flux normalization of 1.6 +/- 0.3_stat +/- 0.8_sys * 10^-11 photons cm^-2 s^-1 at 300 GeV. We also present multiwavelength results taken in the optical (MDM), X-ray (Swift-XRT), and GeV (Fermi-LAT) bands and use these results to construct a broadband spectral energy distribution (SED). Modeling of this SED indicates that homogenous one-zone leptonic scenarios are not adequate to describe emission from the system, with a lepto-hadronic model providing a better fit to the data. ; Comment: Accepted for publication in The Astrophysical Journal