The field-driven reorientation transition of an anisotropic ferromagnetic monolayer is studied within the context of a finite-temperature Green's-function theory. The equilibrium state and the field dependence of the magnon energy gap E-0 are calculated for static magnetic field H applied in plane along an easy or hard axis. In the latter case, the in-plane reorientation of the magnetization is shown to be continuous at T = 0, in agreement with free-spin-wave theory, and discontinuous at finite temperature T > 0, in contrast with the prediction of mean-field theory. The discontinuity in the orientation angle creates a jump in the magnon energy gap, and it is the reason why, for T > 0, the energy does not go to zero at the reorientation field. approach. Above the Curie temperature T-C, the magnon energy gap E-0(H) vanishes for H = 0 in both the easy and hard cases. As H is increased, the gap is found to increase almost linearly with H, but with different slopes depending on the field orientation. In particular, the slope is smaller when H is along the hard axis. Such a magnetic anisotropy of the spin-wave energies is shown to persist well above T-C (T approximate to 1.2T(C)).