  The 'island of inversion' around $^{32}$Mg is one of the most important
paradigm for studying the disappearance of the stabilizing 'magic' of a shell
closure. We present the first Penning-trap mass measurements of the exotic
nuclides $^{29-31}$Na and $^{30-34}$Mg, which allow a precise determination of
the empirical shell gap for $^{32}$Mg. The new value of 1.10(3) MeV is the
lowest observed shell gap for any nuclide with a canonical magic number.
