A multivariate analysis of gamma-ray burst (GRB) bulk properties is presented to discriminate between distinct classes of GRBs. Several variables representing burst duration, fluence, and spectral hardness are considered. Two multivariate clustering procedures are used on a sample of 797 bursts from the Third BATSE Catalog, a nonparametric average linkage hierarchical agglomerative clustering procedure validated with Wilks' Λ* and other multivariate analysis of variance tests and a parametric maximum likelihood model-based clustering procedure assuming multinormal populations calculated with the Expectation-Maximization algorithm and validated with the Bayesian Information Criterion. The two methods yield very similar results. The BATSE GRB population consists of three classes with the following duration/fluence/spectrum bulk properties: class I with long/bright/soft bursts, class II with short/faint/hard bursts, and class III with intermediate/intermediate/soft bursts. One outlier due to spurious data is also present. Classes I and II correspond to those reported by Kouveliotou et al., but class III is clearly defined here for the first time.