We present a reformulation of the Korringa - Kohn - Rostoker (KKR) coherent-potential approximation (CPA) which affords a number of conceptual and practical advantages over conventional formulations of the theory. In particular, as presented here the method can facilitate application to systems that cannot be described properly by a muffin-tin approximation to the cell potential and require a full potential treatment. Also, the formalism allows the derivation of the KKR CPA self-consistency condition within both a scattering matrix and a Green function approach, leading to uniquely defined species-resolved charge densities and densities of states. As shown in a companion paper, this formulation also allows the treatment of the so-called charge-transfer effects associated with Wigner - Seitz cells in an alloy which contain net amounts of charge.