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We present the first set of constant rest-mass ultra-massive oxygen/neon white dwarf cooling tracks with masses larger than 1.29 Msun which fully take into account the effects of general relativity on their structural and evolutionary properties. We have computed the full evolution sequences of 1.29, 1.31, 1.33, 1.35, and 1.369 Msun white dwarfs with the La Plata stellar evolution code, LPCODE. For this work, the standard equations of stellar structure and evolution have been modified to include the full effects of general relativity. For comparison purposes, the same sequences have been computed but for the Newtonian case. According to our calculations, the evolutionary properties of the most massive white dwarfs are strongly modified by general relativity effects. In particular, the resulting stellar radius is markedly smaller in the general relativistic case, being up to 25% smaller than predicted by the Newtonian treatment for the more massive ones. We find that oxygen/neon white dwarfs more massive than 1.369 Msun become gravitationally unstable with respect to general relativity effects. When core chemical distribution due to phase separation on crystallization is considered, such instability occurs at somewhat lower stellar masses, greater than 1.36 Msun. In addition, cooling times for the most massive white dwarf sequences result in about a factor of two smaller than in the Newtonian case at advanced stages of evolution. Finally, a sample of white dwarfs has been identified as ideal candidates to test these general relativistic effects. We conclude that the general relativity effects should be taken into account for an accurate assessment of the structural and evolutionary properties of the most massive white dwarfs.