  We present radio observations of the afterglow of the bright gamma-ray burst
GRB980329 made between one month and several years after the burst, a
re-analysis of previously published submillimeter data, and late-time optical
and near-infrared (NIR) observations of the host galaxy. From the absence of a
spectral break in the optical/NIR colors of the host galaxy, we exclude the
earlier suggestion that GRB980329 lies at a redshift of z >~5. We combine our
data with the numerous multi-wavelength observations of the early afterglow,
fit a comprehensive afterglow model to the entire broadband dataset, and derive
fundamental physical parameters of the blast-wave and its host environment.
Models for which the ejecta expand isotropically require both a high
circumburst density and extreme radiative losses from the shock. No low density
model (n << 10 cm^{-3}) fits the data. A burst with a total energy of ~ 10^{51}
erg, with the ejecta narrowly collimated to an opening angle of a few degrees,
driven into a surrounding medium with density ~ 20 cm^{-3}, provides a
satisfactory fit to the lightcurves over a range of redshifts.
