Gamma rays with energies exceeding 5 ± 1.5 TeV have been detected from Markarian 421 using the Whipple Observatory's 10 m γ-ray telescope. These observations employ a new technique: the so-called large zenith-angle technique. Because they are taken at large zenith angles, the observations yield high statistics data on the multi-TeV part of the spectrum and are well suited for examining the question of a possible energy cutoff. Observations taken during high states on 1995 June 20, 21, and 28 show no evidence for a spectral break. These results conflict with a previous interpretation of the Markarian 421 energy spectrum in which a cutoff due to γ-ray absorption in extragalactic space was postulated.