  We compute the canonical trace of generic determinantal rings and provide a
sufficient condition for the trace to specialize. As an application we
determine the canonical trace $\mbox{tr}(\omega_R)$ of a Cohen-Macaulay ring
$R$ of codimension two, which is generically Gorenstein. It is shown that if
the defining ideal $I$ of $R$ is generated by $n$ elements, then
$\mbox{tr}(\omega_R)$ is generated by the $(n-2)$-minors of the Hilbert-Burch
matrix of $I$.
