The envelope of an elliptical Gaussian complex vector, or equivalently, the amplitude or norm of a bivariate normal random vector has application in many weather and signal processing contexts. We explicitly characterize its distribution in the general case through its probability density, cumulative distribution and moment generating function. Moments and limiting distributions are also derived. These derivations are exploited to also characterize the special cases where the bivariate Gaussian mean vector and covariance matrix have a simpler structure, providing new additional insights in many cases. Simulations illustrate the benefits of using our formulae over Monte Carlo methods. We also use our derivations to get a better initial characterization of the distribution of the observed values in structural Magnetic Resonance Imaging datasets, and of wind speed.