Analysis is presented for conjugate heat transfer in a parallel-plate microchannel. Axial conduction in the fluid and in the adjacent wall is included. The fluid is a constant property gas with a slip-flow velocity distribution. The microchannel is heated by a small region on the channel wall. The analytic solution is given in the form of integrals by the method of Green's functions. Quadrature is used to obtain numerical results for the temperature and heat transfer coefficient on the heated region for various Peclet number, Knudsen number, and wall materials. A region downstream of the heater is also explored. These results have application in the optimal design of small-scale heat transfer devices for biomedical applications, electronic cooling, and advanced fuel cells.