We present spectroscopy and photometry of the He-rich supernova (SN) 2008ax. The early-time spectra show prominent P-Cygni H lines, which decrease with time and disappear completely about 2 months after the explosion. In the same period HeI lines become the most prominent spectral features. SN 2008ax displays the ordinary spectral evolution of a Type IIb supernova. A stringent pre-discovery limit constrains the time of the shock breakout of SN 2008ax to within only a few hours. Its light curve, which peaks in the B band about 20d after the explosion, strongly resembles that of other He-rich core-collapse supernovae. The observed evolution of SN 2008ax is consistent with the explosion of a young Wolf-Rayet (of WNL type) star, which had retained a thin, low-mass shell of its original H envelope. The overall characteristics of SN 2008ax are reminiscent of those of SN 1993J, except for a likely smaller H mass. This may account for the findings that the progenitor of SN 2008ax was a WNL star and not a K supergiant as in the case of SN 1993J, that a prominent early-time peak is missing in the light curve of SN 2008ax, and that H{alpha} is observed at higher velocities in SN 2008ax than in SN 1993J.