Context.AM CVn binaries are potential progenitors of thermonuclear supernovae and strong sources of persistent gravitational wave radiation. For a long time, it has been believed that these systems cannot descend from cataclysmic variables (CVs), at least not in large numbers, because the initial conditions need to be fine-tuned and, even worse, the resulting surface hydrogen abundance would be high enough to be detected which contradicts a defining feature of AM CVn binaries.Aims.Here we show that both claimed weaknesses of the CV formation channel for AM CVn binaries are model-dependent and rely on poorly constrained assumptions for magnetic braking.Methods.We performed binary evolution simulations with the MESA code for different combinations of post-common-envelope white dwarf and companion masses as well as orbital periods assuming the CARB model for strong magnetic braking.Results.We found that AM CVn binaries with extremely-low surface hydrogen abundances are one natural outcome of CV evolution if the donor star has developed a non-negligible helium core prior to the onset of mass transfer. In this case, after hydrogen envelope exhaustion during CV evolution, the donor becomes degenerate and its surface hydrogen abundance substantially drops and becomes undetectable. Our simulations also show that the CV formation channel is able to explain the observed AM CVn binaries with very low mass and bloated donor stars (Gaia14aae and ZTF J1637+49).Conclusions.CVs with evolved donors are likely the progenitors of at least a fraction of AM CVn binaries.