We present high signal-to-noise optical spectra of two luminous super star clusters in the starburst galaxy M82. The data for cluster F and the nearby, highly reddened cluster L were obtained with the William Herschel Telescope (WHT) at a resolution of 1.6A. The blue spectrum (3250-5540A) of cluster F shows features typical of mid-B stars. The red spectra (5730-8790A) 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 6283A diffuse interstellar band appears to be present. The quality of the WHT spectra allows us to considerably improve previous age estimates for cluster F. By comparing the blue spectrum with theoretical model cluster spectra using the PEGASE spectral synthesis code (Fioc & Rocca-Volmerange 1997), we derive an age of 60+/-20 Myr. The strength of the Ca II 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 disk. Cluster F appears to be deep in M82, slightly beyond the main starburst region while the highly obscured cluster L lies near the outer edges of the disk. We derive an absolute V magnitude of -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 approximately 60 Myr ago. ; Comment: 10 pages and 5 figures for publication in MNRAS