  If the theta-null divisor $\Theta_{\rm null}$ is moved to the Prym moduli
space through the diagram
$\mathcal{S}_{g}^{+}\rightarrow\mathcal{M}_{g}\leftarrow\mathcal{R}_{g}$, it
splits into two irreducible components $\mathcal{P}_{\rm\! null}^{+}$ and
$\mathcal{P}_{\rm\! null}^{-}$. Using test curve techniques, we compute the
class of the closure of each of these components in terms of the generating
classes of $\mathrm{Pic}(\overline{\mathcal{R}}_{g})_{\mathbb{Q}}$.
