We present high signal-to-noise ratio optical spectra of two luminous super star clusters in the starburst galaxy M82. The data fur cluster F and the nearby, highly reddened, cluster L were obtained with the William Herschel Telescope (WHT) at a resolution of 1.6 Angstrom. The blue spectrum (3250-5540 Angstrom) of cluster F shows features typical of mid-B stars. The red spectra (5730-8790 Angstrom) of clusters F and L show the Ca II triplet and numerous F- and G-type absorption features. Strong Ca II and Na I interstellar absorption lines arising in M82 are also detected, and the lambda 6283 diffuse interstellar band appears to be present. The quality of the WHT spectra allows us to improve previous age estimates considerably for cluster F. By comparing the blue spectrum with theoretical model cluster spectra using the PEGASE spectral synthesis code, we derive an age of 60 +/- 20 Myr. The strength of the Ca rr triplet is also in accord with this age. Cluster L appears to have a similar age, although this is much less certain. The measured radial velocities for the two clusters differ substantially, indicating that they are located in different regions of the M82 disc. Cluster F appears to he deep in M82, slightly beyond the main starburst region, while the highly obscured cluster L lies near the outer edges of the disc. We derive an absolute magnitude M-V = -16.5 for F indicating that it is an extremely massive cluster. The presence of such a luminous super star cluster suggests that the M82 starburst experienced an episode of intense star formation approximate to 60 Myr ago.