  A central role in recent investigations of the duality of F-theory and
heterotic strings is played by the moduli of principal bundles, with various
structure groups G, over an elliptically fibered Calabi-Yau manifold on which
the heterotic theory is compactified. In this note we propose a simple
algebro-geometric technique for studying the moduli spaces of principal
G-bundles on an arbitrary variety X which is elliptically fibered over a base
S: The moduli space itself is naturally fibered over a weighted projective base
parametrizing spectral covers $\tilde{S}$ of S, and the fibers are identified
as translates of distinguished Pryms of these covers. In nice situations, the
generic Prym fiber is isogenous to the product of a finite group and an abelian
subvariety of $Pic(\tilde{S})$.
