In this study the exact analytical solution of the lumped parameter model of a nonlinear heat transfer process representing the transient convective-radiative cooling of a spherical body has been obtained. The process is governed by a nonlinear ordinary differential equation, and the exact analytical solution has been found in the implicit form of an elementary transcendental function. The obtained exact analytical solution not only yields accurate results but also successfully simulates a recent experimental study of cooling of metallic ball bearings by the combined mechanism of convection and radiation. In addition, the exact explicit solution for a simplified case of the above problem, recently tackled by several researchers in various approximate ways, has also been found. These exact solutions are quite appealing since they are accurate and superior to the available approximate solutions, provide better insight of the physical process, and can also serve as yardsticks for future testing of the approximate solutions.