We present the redshift lower limit of z > or =, slanted 0.6035 for the very high energy (VHE; E > or =, slanted 100 GeV) emitting blazar PKS 1424+240 (PG 1424+240). This limit is inferred from Ly beta and Ly gamma absorption observed in the far-ultraviolet spectra from the Hubble Space Telescope/Cosmic Origins Spectrograph. No VHE-detected blazar has shown solid spectroscopic evidence of being more distant. At this distance, VHE observations by VERITAS are shown to sample historically large gamma-ray opacity values at 500 GeV, extending beyond tau = 4 for low-level models of the extragalactic background light (EBL) and beyond tau = 5 for high levels. The majority of the z = 0.6035 absorption-corrected VHE spectrum appears to exhibit a lower flux than an extrapolation of the contemporaneous Large Area Telescope power-law fit beyond 100 GeV. However, the highest energy VERITAS point is the only point showing agreement with this extrapolation, possibly implying the overestimation of the gamma-ray opacity or the onset of an unexpected VHE spectral feature. A curved log parabola is favored when fitting the full range of gamma-ray data (0.5-500 GeV). While fitting the absorption-corrected VHE data alone results in a harder differential power law than that from the full range, the indices derived using three EBL models are consistent with the physically motivated limit set by Fermi acceleration processes.