  We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$
which are isospectral for the associated classical Dirac operator $D$ must be
isometric. In the case of left-invariant metrics of positive scalar curvature,
we compute and use the smallest eigenvalue of $D^2$. We show analogous results
for left-invariant metrics on $\operatorname{SO}(3)=S^3/\{\pm1\}$ for each of
its two spin structures.
