We use dipole and quadrupole statistics to test the large-scale isotropy of the first 1005 gamma-ray bursts observed by the Burst and Transient Source Experiment (BATSE). In addition to the entire sample of 1005 gamma-ray bursts, many subsets are examined. We use a variety of dipole and quadrupole statistics to search for Galactic and other predicted anisotropies and for anisotropies in a coordinate-system independent manner. We find the gamma-ray burst locations to be consistent with isotropy, e.g., for the total sample the observed Galactic dipole moment differs from the value predicted for isotropy by 0.9 sigma and the observed Galactic quadrupole moment by 0.3 sigma. We estimate for various models the anisotropies that could have been detected. If one-half of the locations were within 86 degrees of the Galactic center, or within 28 degrees of the Galactic plane, the ensuing dipole or quadrupole moment would have typically been detected at the 99% confidence level. We compare the observations with the dipole and quadrupole moments of various Galactic models. Several Galactic gamma-ray bursts models have moments within 2 sigma of the observations; most of the Galactic models proposed to date are no longer in acceptable agreement with the data. Although a spherical Dark Matter Halo distribution could be consistent with the data, the required core radius is larger than the core radius of the Dark Matter Halo used to explain the Galaxy's rotation curve. Gamma-ray bursts are much more isotropic than any observed Galactic population, strongly favoring but not requiring an origin at cosmological distances.