Let $F : M^3 \hookrightarrow {\mathbf R}^5 $ be an embedding of an (oriented) ${\mathbf Z}_2$-homology 3-sphere $M^3$ in ${\mathbf R}^5$. Then $F$ bounds an embedding of an oriented manifold $W^4$ in ${\mathbf R}^5$. It is well known that the signature $\sigma (W^4)$ of $W^4$ is equal to the $\mu$-invariant of $M^3$ modulo 16. In this paper we prove that $\sigma (W^4)$ itself completely determines the regular homotopy class of $F$.