  We establish that if a submanifold $M$ of $\mathbb{R}^n$ is definable in some
o-minimal structure then any definable submanifold $N\subset \mathbb{R}^n$
which is $\mathscr{C}^\infty$ diffeomorphic to $M$, with a diffeomorphism
$h:N\to M$ that is sufficiently close to the identity, must be
$\mathscr{C}^\infty$ definably diffeomorphic to $M$. The definable
diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood
of $M$.
