  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.
