  We define the notion of $\varphi$-perturbation of a densely defined
adjointable mapping and prove that any such mapping $f$ between Hilbert
${\mathcal A}$-modules over a fixed $C^*$-algebra ${\mathcal A}$ with densely
defined corresponding mapping $g$ is ${\mathcal A}$-linear and adjointable in
the classical sense with adjoint $g$. If both $f$ and $g$ are everywhere
defined then they are bounded. Our work concerns with the concept of
Hyers--Ulam--Rassias stability originated from the Th.M. Rassias' stability
theorem that appeared in his paper [On the stability of the linear mapping in
Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297--300]. We also indicate
interesting complementary results in the case where the Hilbert $C^*$-modules
admit non-adjointable $C^*$-linear mappings.
