  The strongest existing constraints on primordial black holes with masses in
the range of $m_{\rm BH} \sim 10^{15}-10^{17} \, {\rm g}$ have been derived
from measurements of the local cosmic-ray electron-positron flux by Voyager 1,
and MeV-scale gamma-ray observations of the Inner Galaxy by COMPTEL and
INTEGRAL. In this paper, we evaluate the sensitivity of future MeV-scale
gamma-ray telescopes such as e-ASTROGAM or AMEGO to Hawking radiation. We show
that such an instrument would be able to provide the strongest constraints on
black holes in the mass range of $m_{\rm BH} \sim (0.6-20) \times 10^{16} \,
{\rm g}$, typically exceeding current constraints by approximately two orders
of magnitude. In scenarios in which the observed 511 keV excess is the result
of Hawking radiation, we find that e-ASTROGAM or AMEGO would not only be able
to detect the Hawking radiation from the Inner Galaxy, but could precisely
measure the abundance and mass distribution of the black holes responsible for
this signal.
