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Ambipolar diffusion is a physical mechanism related to the drift between charged and neutral particles in a partially ionized plasma that is key in many different astrophysical systems. However, understanding its effects is challenging due to basic uncertainties concerning relevant microphysical aspects and the strong constraints it imposes on the numerical modeling. Our aim is to introduce a numerical tool that allows us to address complex problems involving ambipolar diffusion in which, additionally, departures from ionization equilibrium are important or high resolution is needed. The primary application of this tool is for solar atmosphere calculations, but the methods and results presented here may also have a potential impact on other astrophysical systems. We have developed a new module for the stellar atmosphere Bifrost code that improves its computational capabilities of the ambipolar diffusion term in the Generalized Ohm's Law. This module includes, among other things, collision terms adequate to processes in the coolest regions in the solar chromosphere. As a key feature of the module, we have implemented the Super Time-Stepping (STS) technique, that allows an important acceleration of the calculations. We have also introduced hyperdiffusion terms to guarantee the stability of the code. We show that to have an accurate value for the ambipolar diffusion coefficient in the solar atmosphere it is necessary to include as atomic elements in the equation of state not only hydrogen and helium but also the main electron donors like sodium, silicon and potassium. In addition, we establish a range of criteria to set up an automatic selection of the free parameters of the STS method that guarantees the best performance, optimizing the stability and speed for the ambipolar diffusion calculations. We validate the STS implementation by comparison with a self-similar analytical solution.