  As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules,
we show in this article that the module structure and the orthogonality
structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module
structure. In fact, we have a more general result as follows. Let $A$ be a
$C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of
$A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an
$A$-module map, not assumed to be bounded but satisfying $$ \langle
\Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\
0, $$ then there exists a unique central positive multiplier $u\in M(I_E)$ such
that $$ \langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad
(x,y\in E). $$ As a consequence, $\Phi$ is automatically bounded, the induced
map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$
is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition,
$\Phi$ is bijective, then $E$ is isomorphic to $F$.
