<p>A novel analytical solution has been formulated and obtained for the first time to the second initial boundary-value problem on conjugate heat transfer between two plates with the anisotropy of general form in both plates. The generality of anisotropy means not only the presence of nonzero components of the thermal-conductivity tensors of the plates, but also different orientations of the principal axes of the thermal-conductivity tensors; here, extra-diagonal coefficients of the thermal-conductivity tensor may have arbitrary signs. Since the differential equations of anisotropic thermal conductivity in both plates contain mixed derivatives of temperature with respect to space variables, boundary conjugation conditions include all the components of the temperature gradient, even of the boundary is plane. For this reason, the method of separation of variables would not do for solving equations with mixed derivatives. Therefore, we have to use methods of Fourier and Laplace integral transformations accordingly with respect to space variable and with respect to time. Here, in Fourier transforms, in the terms in the differential heat-conduction equations containing mixed derivatives, there arise imaginary coefficients at the extradiagonal coefficients of the thermal-conductivity tensor.</p>