  We present new H-band echelle spectra, obtained with the NIRSPEC spectrograph
at Keck II, for the massive star cluster "B" in the nearby dwarf irregular
galaxy NGC 1569. From spectral synthesis and equivalent width measurements we
obtain abundances and abundance patterns. We derive an Fe abundance of
[Fe/H]=-0.63+/-0.08, a super-solar [alpha/Fe] abundance ratio of +0.31+/-0.09,
and an O abundance of [O/H]=-0.29+/-0.07. We also measure a low 12C/13C = 5+/-1
isotopic ratio. Using archival imaging from the Advanced Camera for Surveys on
board HST, we construct a colour-magnitude diagram (CMD) for the cluster in
which we identify about 60 red supergiant (RSG) stars, consistent with the
strong RSG features seen in the H-band spectrum. The mean effective temperature
of these RSGs, derived from their observed colours and weighted by their
estimated H-band luminosities, is 3790 K, in excellent agreement with our
spectroscopic estimate of Teff = 3800+/-200 K. From the CMD we derive an age of
15-25 Myr, slightly older than previous estimates based on integrated
broad-band colours. We derive a radial velocity of -78+/-3 km/s and a velocity
dispersion of 9.6+/-0.3 km/s. In combination with an estimate of the half-light
radius of 0.20"+/-0.05" from the HST data, this leads to a dynamical mass of
(4.4+/-1.1)E5 Msun. The dynamical mass agrees very well with the mass predicted
by simple stellar population models for a cluster of this age and luminosity,
assuming a normal stellar IMF. The cluster core radius appears smaller at
longer wavelengths, as has previously been found in other extragalactic young
star clusters.
