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Multilayered, counterflow, parallel-plate heat exchangers are studied theoretically and numerically. The analysis, carried out for constant property fluids, assumes a fully developed laminar flow with moderately large Peclet numbers in the flow channels, so that longitudinal conduction can be neglected in the fluids in first approximation. By way of contrast, both axial and transverse conduction effects are fully accounted for in the heat conducting plates, making the analysis relevant to the design of mini- and micro-heat exchangers. The exact solution for the temperature field is obtained in terms of eigenfunction expansions involving infinite sets of both positive and negative eigenvalues. Based on previous results, the eigenfunctions are expressed in terms of Whittaker functions, leading to an analytical eigencondition that provides the eigenvalues numerically. Making use of a newly defined orthogonality condition, the expansion coefficients are determined through an infinite system of linear equations that must be truncated to a sufficiently large number of terms to obtain a numerical solution of the problem. The exact solution for the temperature field provides analytical expressions for the interfacial and bulk temperatures, local heat-transfer rates, overall heat-transfer coefficient, Nusselt numbers, and outlet bulk temperatures of both fluids. Numerical results evaluated using a moderate number of terms in the series are compared against finite difference solutions of the problem and against previous results reported in the literature, showing excellent agreement in both cases. The exact solution presented here may be used as a benchmark case for computational heat transfer codes, and enables future spectral and parametric studies of the problem.