We present radio observations of the afterglow of the bright γ-ray burst GRB 980329 made between 1 month and several years after the burst, a reanalysis of previously published submillimeter data, and latetime 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 GRB 980329 lies at a redshift of z ≳ 5. We combine our data with the numerous multiwavelength observations of the early afterglow, fit a comprehensive afterglow model to the entire broadband data set, 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 ∼1051 ergs, with the ejecta narrowly collimated to an opening angle of a few degrees, driven into a surrounding medium with density of ∼20 cm-3, provides a satisfactory fit to the light curves over a range of redshifts.