  We present infrared spectroscopy of the Antennae Galaxies (NGC 4038/4039)
with NIRSPEC at the W. M. Keck Observatory. We imaged the star clusters in the
vicinity of the southern nucleus (NGC 4039) in 0.39" seeing in K-band using
NIRSPEC's slit-viewing camera. The brightest star cluster revealed in the
near-IR (M_K(0) = -17.9) is insignificant optically, but coincident with the
highest surface brightness peak in the mid-IR (12-18 um) ISO image presented by
Mirabel et al (1998). We obtained high signal-to-noise 2.03-2.45 um spectra of
the nucleus and the obscured star cluster at R = 1900.
  The cluster is very young (age ~ 4 Myr), massive (M ~ 16E6 M_sun), and
compact (density ~ 115 M_sun pc^(-3) within a 32 pc half-light radius),
assuming a Salpeter IMF (0.1-100 M_sun). Its hot stars have a radiation field
characterized by T_eff ~ 39,000 K, and they ionize a compact HII region with
n_e ~ 10^4 cm^(-3). The stars are deeply embedded in gas and dust (A_V = 9-10
mag), and their strong FUV field powers a clumpy photodissociation region with
densities n_H > 10^5 cm^(-3) on scales of ~ 200 pc, radiating L{H_2 1-0 S(1)}=
9600 L_sun.
