The $\Xi_c^0$ baryon is unstable and usually decays into charmless final states by the $c \to s u\overline{d}$ transition. It can, however, also disintegrate into a $\pi^-$ meson and a $\Lambda_c^+$ baryon via $s$ quark decay or via $cs\to d c$ weak scattering. The interplay between the latter two processes governs the size of the branching fraction ${\cal{B}}$$(\Xi_c^0\to \pi^-\Lambda_c^+)$, first measured here to be $(0.55\pm 0.02 \pm 0.18)$%, where the first uncertainty is statistical and second systematic. This result is compatible with the larger of the theoretical predictions that connect models of hyperon decays using partially conserved axial currents and SU(3) symmetry with those involving the heavy-quark expansion and heavy-quark symmetry. In addition, the branching fraction of the normalization channel, ${\cal{B}}(\Xi_c^+\to p K^- \pi^+) = (1.135 \pm 0.002 \pm 0.387)$% is measured.