  Let $F$ be a non-Archimedean local field of residual characteristic $p$, and
$\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence,
between irreducible, $n$-dimensional, smooth representations of the Weil group
of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use
an explicit description of the correspondence from an earlier paper, and
otherwise entirely elementary methods, to show that it respects the
relationship of congruence modulo $\ell$. The $\ell$-modular correspondence
thereby becomes as effective as the complex one.
