Sharp-interface and thin-interface asymptotic analyses are presented for a generalization of the Beckermann Karma phase-field model for solidification of a dilute binary alloy when the interface curvature is macroscopic. The ratio of diffusivities, R-m equivalent to D-s(')/D-m('), in the solid and melt is arbitrary with 0 <= R-m <= 1. Discrepancies between this diffuse-interface model and the classical, two sided solutal model (TSM) description are quantified up to second order in the small parameter e that controls the interface thickness. We uncover extra terms in the interface species flux balance and in the Gibbs-Thomson equilibrium condition introduced by the finite width of the interface. Asymptotic results in the limit of rapid-interfacial kinetics are presented for both finite phase-field mobility and a quasi-steady state approximation for the phase-field wherein the phase-field responds passively to the concentration field. The possibility of adding additional terms to the phase-field version of the species conservation equation is explored as a means of achieving O(epsilon(2)) consistency with the classical model. Our results naturally lead to a generalization of the anti-trapping solutal flux suggested by Karma [Phase-field-formulation for quantitative modeling of alloy solidification, Phys. Rev. Lett. 87(11) (2001) 115701] for the limit R-m = 0. Achieving second order accuracy for arbitrary Rm requires judicious choices for the interpolating functions; these are calculated a posteriori using the functional forms of the error terms as a guide. (c) 2006 Elsevier B.V. All rights reserved.