  We study by molecular dynamics the interplay between arrest and
crystallization in hard spheres. For state points in the plane of volume
fraction ($0.54 \leq phi \leq 0.63$) and polydispersity ($0 \leq s \leq
0.085$), we delineate states that spontaneously crystallize from those that do
not. For noncrystallizing (or precrystallization) samples we find
isodiffusivity lines consistent with an ideal glass transition at $\phi_g
\approx 0.585$, independent of $s$. Despite this, for $s<0.05$, crystallization
occurs at $\phi > \phi_g$. This happens on time scales for which the system is
aging, and a diffusive regime in the mean square displacement is not reached;
by those criteria, the system is a glass. Hence, contrary to a widespread
assumption in the colloid literature, the occurrence of spontaneous
crystallization within a bulk amorphous state does not prove that this state
was an ergodic fluid rather than a glass.
