  Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye
congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai
partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in
$\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and
$Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using
mirror symmetry.
