  The nature of the bipolar, $\gamma$-ray Fermi bubbles (FB) is still unclear,
in part because their faint, high-latitude X-ray counterpart has until now
eluded a clear detection. We stack ROSAT data at varying distances from the FB
edges, thus boosting the signal and identifying an expanding shell behind the
southwest, southeast, and northwest edges, albeit not in the dusty northeast
sector near Loop I. A Primakoff-like model for the underlying flow is invoked
to show that the signals are consistent with halo gas heated by a strong,
forward shock to $\sim$keV temperatures. Assuming ion--electron thermal
equilibrium then implies a $\sim10^{56}$ erg event near the Galactic centre
$\sim7$ Myr ago. However, the reported high absorption-line velocities suggest
a preferential shock-heating of ions, and thus more energetic ($\sim 10^{57}$
erg), younger ($\lesssim 3$ Myr) FBs.
