A conjugate heat transfer problem of a second-grade viscoelastic fluid past a plate fin was studied. Governing equations, including heat conduction equation of a fin, and continuity equation, momentum equation and energy equation of a second-grade fluid, were analyzed by a combination of a series expansion method, the similarity transformation and a second-order accurate finite-difference method. Solutions of a stagnation flow (beta = 1.0) at the an tip and a flat-plate how (beta = 0) on the fin surface were obtained by a generalized Falkner-Skan flow derivation. These solutions were used to iterate with the heat conduction equation of the fin to obtain distributions of the local convective heat transfer coefficient and the fin temperature. Ranges of dimensionless parameters, the Prandtl number (Pr), the elastic number (E) and the conduction-convection coefficient (N-alpha) are from 0.1 to 100, 0.001 to 0.3, and 0.5 to 2.0, respectively. Results indicated that elastic effect in the flow can increase the local heat transfer coefficient and enhance the heat transfer of a fin. Also, same as results from Newtonian fluid flow and conduction analysis of a fin, a better heat transfer is obtained with a larger N-proportional to and Pr.