  We study the Hopfield model with pure $p$-spin interactions with even $p\geq
4$, and a number of patterns, M(N) growing with the system size, $N$, as $M(N)
= \a N^{p-1}$. We prove the existence of a critical temperature $\b_p$
characterized as the first time quenched and annealed free energy differ. We
prove that as $p\uparrow\infty$, $\b_p\to\sqrt {\a 2\ln 2}$. Moreover, we show
that for any $\a>0$ and for all inverse temperatures $\b$, the free energy
converges to that of the REM at inverse temperature $\b/\sqrt\a$. Moreover,
above the critical temperature the distribution of the replica overlap is
concentrated at zero. We show that for large enough $\a$, there exists a
non-empty interval of in the low temperature regime where the distribution has
mass both near zero and near $\pm 1$. As was first shown by M. Talagrand in the
case of the $p$-spin SK model, this implies the the Gibbs measure at low
temperatures is concentrated, asymptotically for large $N$, on a countable
union of disjoint sets, no finite subset of which has full mass. Finally, we
show that there is $\a_p\sim 1/p!$ such that for $\a>\a_p$ the set carrying
almost all mass does not contain the original patterns. In this sense we
describe a genuine spin glass transition.
  Our approach follows that of Talagrand's analysis of the $p$-spin SK-model.
The more complex structure of the random interactions necessitates, however,
considerable technical modifications. In particular, various results that
follow easily in the Gaussian case from integration by parts fromulas have to
be derived by expansion techniques.
