  Rare two-body decays of the top quark into a neutral bottom-quark meson plus
an up- or charm-quark: $t\to {\overline B}^0+ u, c$; $t\to {\overline B}^0_{s}+
c,u$; and $t \to \Upsilon(nS)+ c,u$, are studied for the first time. The
corresponding partials widths are computed at leading order in the
non-relativistic QCD framework. The sums of all two-body branching ratios
amount to $\mathcal{B}(t \to {\overline B}^0+ {\rm jet}) \approx \mathcal{B}(t
\to {\overline B}^0_{s}+ {\rm jet}) \approx 4.2\cdot 10^{-5}$ and
$\mathcal{B}(t \to \Upsilon(nS)+ {\rm jet}) \approx 2\cdot 10^{-9}$,
respectively. The feasibility to observe the $t\to {\overline B}^0_{(s)}+{\rm
jet}$ decay is estimated in top-pair events produced in proton-proton
collisions at $\sqrt{s} = 14, 100$ TeV at the LHC and FCC, respectively.
Combining many exclusive hadronic ${\overline B}^0_{(s)}$ decays, with $J/\psi$
or $D^{0,\pm}$ final states, about 50 (16000) events are expected in 3 (20)
ab$^{-1}$ of integrated luminosity at the LHC (FCC), after typical selection
criteria, acceptance, and efficiency losses. An observation of the two-body
top-quark decay can also be achieved in the interesting $t\to
b(\rm{jet})+c(\rm{jet})$ dijet final state, where the ${\overline B}^0_{(s)}$
decay products are reconstructed as a jet, with 5300 and 1.4 million signal
events above backgrounds expected after selection criteria at the LHC and FCC,
respectively. Such unique final states provide a new direct method to precisely
measure the top-quark mass via simple 2-body invariant mass analyses.
