Let $A$ and $B$ be two Fréchet locally $C^*$-algebras, let $E$ be a full Hilbert $A$-module, and let $F$ be a Hilbert $B$-module. We show that a bijective linear map $\Phi\colon E\to F$ is a unitary operator from $E$ to $F$ if and only if there is a map $\varphi\colon A\to B$ with closed range such that $\Phi(\xi a)=\Phi(\xi)\varphi(a)$ and $\varphi(\langle\xi,\eta\rangle)=\langle\Phi(\xi),\Phi(\eta)\rangle$ for all $a\in A$ and for all $\xi,\eta\in E$.