The non-linear equations called the Stenflo equations are usually used for the analytical description of the propagation of internal gravity waves in the Earth’s upper atmosphere. The solutions in the form of dipole vortices, tripole vortices and vortex chains were previously obtained with the help of these equations. The Stenflo equations also describe rogue waves, breathers, and dark solitons. It is known that when disturbances cease to be small, their profiles are usually deformed and it is assumed that they cannot be represented in the form of plane waves. This paper shows that this is not always the case for internal gravity waves and that even at large amplitudes these waves can propagate as plane waves. An exact solution of the system of nonlinear Stenflo equations for internal gravity waves containing nonlinear terms in the form of Poisson brackets is presented. The solution is obtained in the form of plane waves with an arbitrary amplitude. To find a solution, the original system of equations was transformed. We split it into equations for the stream functions and vorticity functions, as well as equations for the perturbed density. To solve the obtained equations, the procedure of successive zeroing of Poisson brackets was applied. As a result, linear equations were obtained, that allow finding the accurate analytical solutions for internal gravity waves in the form of plane waves with arbitrary amplitude. By solving these linear equations in two different ways, we analytically found expressions for the perturbed quantities and the dispersion equation. The nonlinear equations we obtained for the current, vorticity, and perturbed density functions can be used to find other nonlinear solutions. The presented solutions in the form of plane waves with an arbitrary amplitude may be of interest for the analysis of the propagation of internal gravity waves in the Earth’s atmosphere and the interpretation of experimental data.