Effective design of autopilots for fixed-wing unmanned aerial vehicles (UAVs) is still a great challenge, due to unmodeled effects and uncertainties that these vehicles exhibit during flight. Unmodeled effects and uncertainties comprise longitudinal/lateral cross-couplings, as well as poor knowledge of equilibrium points (trimming points) of the UAV dynamics. The main contribution of this article is a new adaptive autopilot design, based on uncertain Euler-Lagrange dynamics of the UAV and where the control can explicitly take into account under-actuation in the dynamics, reduced structural knowledge of cross-couplings and trimming points. This system uncertainty is handled via appropriately designed adaptive laws: stability of the controlled UAV is analyzed. Hardware-in-the-loop tests, comparisons with an Ardupilot autopilot and with a robustified autopilot validate the effectiveness of the control design, even in the presence of strong saturation of the UAV actuators.