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Absolute frequency measurements on single trapped ions reached tremendous accuracies by probing narrow transitions in tightly confined ions (resolved sideband regime). A variety of interesting transitions may hardly be studied in this regime, because their transition linewidths well exceed common (secular) trap frequencies. Accurate spectroscopy in the so-called weak binding limit is challenging, since an interrogating laser induces detuning dependent heating and cooling which distorts the line profile. We present both theory and experimental demonstration of a spectroscopic method which essentially removes these limitations and allows us to observe a well understood line shape with high signal-to-noise ratio. We measured the 24Mg and 26Mg component of the 3s-3p fine structure doublet in Mg+. Both lines played an decisive role in the Many Multiplet studies of quasar absorption spectra suggesting a smaller fine-structure constant in early epochs of the universe and where requested for isotopically resolved re-calibration. Our measurement provides absolute transition frequencies, isotope shifts and fine structure splittings that exceed the accuracy of previous literature values by more than two orders of magnitude.