We report on the VERITAS discovery of very high energy (VHE) gamma-ray emission above 200 GeV from the high-frequency-peaked BL Lac (HBL) object RX J0648.7+1516 (GB J0648+1516), associated with 1FGL J0648.8+1516. The photon spectrum above 200 GeV is fitted by a power law dN/dE = F ₀(E/E ₀)^–Γ with a photon index Γ of 4.4 ± 0.8_stat ± 0.3_syst and a flux normalization F ₀ of (2.3 ± 0.5_stat ± 1.2_sys) × 10^–11 TeV^–1 cm^–2 s^–1 with E ₀ = 300 GeV. No VHE variability is detected during VERITAS observations of RX J0648.7+1516 between 2010 March 4 and April 15. Following the VHE discovery, the optical identification and spectroscopic redshift were obtained using the Shane 3 m Telescope at the Lick Observatory, showing the unidentified object to be a BL Lac type with a redshift of z = 0.179. Broadband multiwavelength observations contemporaneous with the VERITAS exposure period can be used to subclassify the blazar as an HBL object, including data from the MDM observatory, Swift-UVOT, and X-Ray Telescope, and continuous monitoring at photon energies above 1 GeV from the FermiLarge Area Telescope (LAT). We find that in the absence of undetected, high-energy rapid variability, the one-zone synchrotron self-Compton (SSC) model overproduces the high-energy gamma-ray emission measured by the Fermi-LAT over 2.3 years. The spectral energy distribution can be parameterized satisfactorily with an external-Compton or lepto-hadronic model, which have two and six additional free parameters, respectively, compared to the one-zone SSC model.