We present optical spectroscopy obtained with the Space Telescope Imaging Spectrograph (STIS) of five young massive star clusters in the starburst galaxy M82. A detailed analysis is performed for one cluster `M82-A1 and its immediate environment in the starburst core. From HST archive images, we find that it is elliptical with an effective radius of 3.0+/-0.5 pc and is surrounded by a compact (r=4.5+/-0.5 pc) H II region. We determine the age and reddening of M82-A1 using synthetic spectra from population synthesis models by fitting both the continuum energy distribution and the depth of the Balmer jump. We find an age of 6.4+/-0.5 Myr and a photometric mass estimate of M=7-13 x 10^5 solar masses. We associate its formation with the most recent starburst event 4-6 Myr ago. We find that the oxygen abundance of the H II region surrounding M82-A1 is solar or slightly higher. The H II region has a high pressure P/k = 1-2 x 10^7 cm^-3 K. The diffuse gas in region A has a slightly lower pressure, which together with the broad H alpha emission line width, suggests that both the thermal and turbulent pressures in the M82 starburst core are unusually high. We discuss how this environment has affected the evolution of the cluster wind for M82-A1. We find that the high pressure may have caused the pressure-driven bubble to stall. We also obtain spectroscopic ages for clusters B1-2 and B2-1 in the `fossil starburst region and for the intermediate age clusters F and L. These are consistent with earlier studies and demonstrate that star formation activity, sufficiently intense to produce super star clusters, has been going on in M82 during the past Gyr, perhaps in discrete and localized episodes.
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