  In the minimal $U(1)$ extension of the Standard Model (SM), a new gauge boson
referred to as "dark photon" is predicted. The dark-photon mass can be
generated from an additional Higgs mechanism associated with a dark scalar
boson. At $B$-factories such as Belle II, large numbers of $B$-mesons are
produced and can decay to a kaon plus the dark scalar via the latter's mixing
with the SM Higgs boson. We evaluate the sensitivity of Belle II for the case
in which the dark scalar decays exclusively into a pair of dark photons via the
new $U(1)$ gauge coupling, and the dark photons are long lived owing to a small
kinetic mixing $\epsilon$. We study the experimental signature in which each
dark photon decays into a pair of charged leptons, pions, or kaons, resulting
in a pair of displaced vertices, and argue that the search is essentially
background-free. We perform detailed Monte-Carlo simulations to determine the
expected number of signal events at Belle II with an integrated luminosity of
50 ab$^{-1}$, taking into account the efficiencies for both
final-state-particle identification and displaced tracking. We find that for
experimentally allowed values of the scalar mixing angle and kinematically
allowed dark-photon and dark-scalar masses, the proposed search is uniquely
sensitive to the medium-$\epsilon$ regime, which is currently mostly unexcluded
by experiments.
