Let $M$ be a compact Riemannian manifold and $^1{\rm Spec}(M)$ be the spectrum of the Laplacian acting on 1-forms. It is well known that the Veronese surface is the only compact minimal surface of $S^4$ with scalar curvature $r=\frac23$ (equivalently, $\|B\|^2=\frac43$, where $B$ is the second fundamental form) [S. S. Chern, M. do Carmo\ and\ S. Kobayashi, in {\it Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968)}, 59--75, Springer, New York, 1970; MR0273546 (42 \#8424)]. In this paper the authors give the following theorem: Let $M$ be a compact minimal surface, and $M'$ be the Veronese surface, both in $S^4$. If the normal connection of $M$ is flat and $^1{\rm Spec}(M)={}^1{\rm Spec}(M')$, then $M$ coincides with $M'$. \par \{Reviewer's remark: The reviewer believes that this result follows, without the assumption of flatness of the normal connection of $M$, from Theorem A ((i) for $m=2$) of [S. Tanno, Proc. Amer. Math. Soc. {\bf 45} (1974), 125--129; MR0343321 (49 \#8063)].\}