Abstract: We show that a bi-invariant metric on a compact connected Lie group G is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric g_0 on G , there is a positive integer N such that, within a neighborhood of g_0 in the class of left-invariant metrics of at most the same volume, g_0 is uniquely determined by the first N distinct nonzero eigenvalues of its Laplacian (ignoring multiplicities). In the case where G is simple, N can be chosen to be 2 . This is joint work with Dorothee Schueth and Craig Sutton.